Space Cruiser Yamato has the Earth see the approach of the Comet Empire, even though it's light years away and the light from it wouldn't reach us yet. Also, the Comet Empire is the size of a small planet - big, but not big enough to be seen at that distance anyway.
In the Americanization, Star Blazers, the newly launched Argo makes its first hyperspace jumps, traveling light years from Earth to Mars. Must have been a bit of a detour involved.
In Tengen Toppa Gurren Lagann, the final battle against the Anti-Spiral and the eponymous mech takes place on a scale so huge entire galaxies are visible in single shots, and any movement is far beyond light-speed. By that point in the series, physics have been ground to a fine powder.
At least in the movie, the final battle is mentioned to take place in a pocket universe that can be shaped with Spiral Power, so the laws of physics as we know them(and scale, for that matter, seeing how you have a mecha with a visible planet on its head throwing a galaxy at another mecha while standing on a much bigger galaxy) need not to apply.
In Dragon Ball Z, Goku learns to teleport himself in such a manner that he moves at the speed of light (In the dub, he mentions the exact speed). He states that he could have used this technique to get to Earth faster than with his ship, if he thought it was necessary. He was traveling on that ship from a planet millions of light years away. That trip took 3 years. In short, his teleport ability is millions of times slower than his ship was.
This is a Dub-Induced Plot Hole. Instant Transmission is exactly what it sounds like: instantaneous. There was none of this "speed of light" silliness in the original script.
Any time you are dealing with a speedster like The Flash, the laws of drama beat up the laws of physics and take their lunch money.
The Flash's writers eventually gave up trying to apply science to the Scarlet Speedster, and introduced the magical Speed Force.
Even given the assumption that the Flash can move at an arbitrarily high velocity without any unintended side effects, most of his adventures still don't make any sense, because otherwise he would be able to defeat any villain before they had time to react. The laws of drama make him absurdly fast at some times and absurdly slow at others.
Films — Live-Action
In Deep Impact, the spaceship in Earth orbit maneuvers down a fissure in the asteroid to detonate at the asteroid's core. The ship starts off (to a good enough approximation compared to the other velocities) stationary relative to the Earth, so the ship/asteroid relative velocity is similar to the Earth/asteroid relative velocity. The Earth/asteroid relative velocity can't be less than the Earth escape velocity (11 km∕s) and is more likely on the order of the Earth's orbital velocity (30 km∕s). So they're navigating down that narrow fissure at something between 10 and 40 km∕s....
Then again, the space ship in question did have an Orion Drive, which can produce an insane amount of thrust.
Starship Troopers once again shines through with a huge asteroid passing a battleship vastly distant from Earth, with the same asteroid striking the Earth mere hours later. To add insult to injury, the battleship was also moving towards the asteroid, yet the asteroid seems to slowly pass by happily ripping off a section of the ship as it does so - suggesting the ship is, in fact, flying backwards at a velocity very close to but slightly slower than the asteroid.
Might be handwaved by saying that interstellar travel in this universe works by distorting space around the travelling object such that lightyears are compressed to kilometers. Then speeds of a few km∕h would be sufficient...
Unfortunately, at the time the ship was orbiting Jupiter, not engaging in interstellar travel. And travelling from Jupiter to Earth at speeds on the order of orbiting spacecraft (10s of kilometers per second) takes years.
Also, when he said "Light speed too slow?!" it didn't sound like a joke, like an obvious idiot saying something like that when of course light speed is nothing when it comes to interstellar travel; the situation is that they were chasing a particularly fast ship and the punchline is pretty much that there's a setting called Ludicrous Speed that takes you from stars zipping by to plaid.
In just every work involving an Asteroid Thicket, spaceships are seen passing rocks at relative speeds that would be about right for seagoing vessels passing an island. If there is a chase inside this asteroid field, it looks like a downtown car chase (or jetplane chase, if you're generous). In reality, even with today's technology, flybys are measured in dozens of kilometers per second.
In Star Trek: The Motion Picture the Enterprise has left space dock and is on its way to intercept V'Ger. It heads away from Earth and, moving at only sublight speed, manages to pass Jupiter only a few moments after leaving Earth orbit.
In The Empire Strikes Back, near the end, the characters are looking out the side of a ship while watching some stellar object. The object has been identified by various sources as the Galaxy. However, if this was true, then given that the stellar object was visibly spinning, the Galaxy must therefore be spinning faster than the speed of light. The 2004 Blu-Ray commentary has either retconned or clarified that the object in question was a stellar nebula instead.
Unfortunately, that isn't really better in the sense of no longer defying the expected laws of physics. A nebula is still too large to rotate visibly and still be moving at less than lightspeed, and has the additional problem of not having enough mass for gravity to hold it together as it rotates.
This is a prime example of Rule of Drama in action. The filmmakers knew perfectly well that in reality, the galaxy would seem to be utterly static. But on screen, a still image looked like a painting, and ruined the feel of the shot.
Star Trek Into Darkness: Two ships battle in warp space, one knocking the other out. At this point they are 250,000 miles from Earth, as pointed out by dialogue. This is really close to the Earth for either ship to have still been in warp, even battling the way they were. One of their computers would have been alerting the crew that they were near home. It's kind of like driving at 100 miles per hour and stopping on a dime an inch from your house.
In the Soviet two-part film Moscow — Cassiopeia, the ZARYa is supposed to be a relativistic ship crewed by teens whose journey is expected to take decades even from their viewpoint. And yet, in a matter of hours, the ship gets close to Proxima Centauri, which is about 4.24 lightyears away. Given that they engage in real-time communication with Earth, their speed can't be a high percentage of the speed of light, meaning it should take many years before they can get close to another star. Then again, the movie isn't really trying to be hard sci-fi, given that IOO pretty much breaks every law of physics we know.
Justified in Animorphs where travel through Zero Space is explicitly stated to be relatively random, where the same distance can take either hours or months, depending on how Zero Space shifted.
In Tunnel in the Sky, two teenagers notice a new visible star above the alien world they're stranded on, and conclude that they've just witnessed a nova. At the book's end, it's revealed a nova is what interfered with their Cool Gate back to Earth. If it's the same nova — which is strongly implied; indeed, the chapter where they see it is titled "The Nova", and it reads like a Chekhov's Gun — then the boys shouldn't have been able to see its light until years after it happened.
Overlapping with Distance, there's an essay out there that analyzes the Threadfighting tactics in the Dragonriders of Pern stories. It concludes that in order for the tactics described to work against the Threadfall patterns described, either the dragons must be flying at barely-subsonic velocities, Thread is drifting downward somewhat slower than a falling leaf, or the dragons are emitting gouts of flame better than half a mile long.
In E. E. “Doc” Smith's The Skylark of Space, the titular Skylark accelerates to 3 times the speed of light in the space of 20 minutes, knocking its inhabitants unconscious. While Smith acknowledges that this violates Einstein's universal speed limit, he fails to acknowledge what such an extreme acceleration (about 75,000 G's) would do to a human crew.
It's possible he realised this error, because the villains of Skylark Three, the Fenachrone, have found a way to make their stardrive operate simultaneously and equally on everything within the ship, thus avoiding this conundrum.
Star Carrier: Earth Strike describes the railgun cruiser Kinkaid's spinal mount as accelerating a slug at 500 gravities down a 1 kilometer barrel. This results in a muzzle velocity of about 3.1 kilometers per second, only about four times the muzzle velocity of the average Real Life assault rifle. At that speed it would take a very large, very heavy slug to do much damage (at least, damage noticeable to ships that regularly weather hits by 10 kiloton nukes), which sort of defeats the purpose of using a railgun that long (real railguns have achieved 2.4 km/s with a much shorter barrel).
Live Action TV
The USS Enterprise in Star Trek: The Original Series traveled to the edge of the galaxy (in "Where No Man Has Gone Before" and "By Any Other Name") and to the center of the galaxy (in Star Trek V) in the space of a single episode. The trip from one galaxy to another would take about 300 years (though the trip would be made with the modifications of extra-galactic aliens who had engines that were better than the Federation's). Yet in Star Trek: Voyager, when ships were about 2-4 times faster, the estimated travel time to Earth from the opposite side of the galaxy was upwards of 70 years.
Several episodes of original Trek have the Enterprise departing the planet-of-the-week at Warp Factor 1. This would mean they're travelling at the speed of light. At that rate, it would take them years just to get to the nearest neighboring star system. (Perhaps Kirk cranks it up to warp 6 once they're past the asteropause.)
Many episodes of Voyager also have the ship traveling at sublight for extended periods of time. This is no particular reason for a ship that's trying to get home in under 7 years to travel at sublight. In fact, they may as well be standing still (to any stellar body nearby).
They need to get supplies, refuel, etc., and have to drop out of warp to do it.
By the time of Star Trek: The Next Generation, a Warp Speed scale was firmly established by Paramount, where a speed of Warp X (below Warp 9) meant the ship was travelling at X^3.3333... times the speed of light. This makes Warp 1 equal to light speed, Warp 2 just a hair over 10 times the speed of light, and Warp 9 a little more than 1500 times light speed. Yet in the episode "Where Silence Has Lease", the Enterprise traverses the 1.3 parsec distance to the edge of a giant space cloud at Warp 2 in about 30 seconds.
Which fits with the Voyager example, which has the cruising speed of the ship at about 1,000 times the speed of light.
...and which was completely abandoned by Deep Space Nine, where traveling anywhere invariably took about half a day. Runabouts (established as having a top speed of warp five) seemed to be able to reach Earth, Cardassia, and various other locations in the same short amount of time (while the distances aren't given, that would put both the Federation and Cardassian capitals within two light years of each other at most). On another occasion, a runabout travels to a planet given as five light years away in a few hours (it should take over a week).
On one occasion, the Enterprise-D's warp drive has to be shut down for some kind of procedure, and they keep on pushing toward their destination at impulse speeds in the meantime. On an interstellar scale, you're not going to cover any meaningful distance at sublight speeds; they might as well have stopped altogether.
Depending on the Writer and how badly they feel like contradicting themselves some more, "impulse" might mean anything from real-world rocket engines to a weaker secondary FTL drive (although one possible handwave is that it may be a generic term in-universe). Some episodes have ships travelling at FTL speeds on impulse power without explanation.
In Star Trek: Enterprise, the Klingon homeworld is several days' travel from Earth, which would put the two empires right on top of each other, given the increases in cruising speed in the other incarnations of the franchise. (Though one interesting exception: in the first episode of Star Trek: Enterprise, Trip describes the ship's top speed in terms of how long it would take to travel to Jupiter and back, and he's exactly right, based on conventional estimates of how warp factors work). In fact if you use those same calculations, the Klingon home planet would be two and a half lightyears from Earth; the nearest star to us in Real Life is four and a half. Missed it by that much. Which is, really, still a big distance in conventional terms, but isn't that much in terms of space.
Star Trek: The Motion Picture started the tradition of showing stars streaming past the window whenever they were at warp speed. Even at the movie-Enterprise's maximum safe cruising speed of warp 6 (TOS scale), they'd still only be going 216 times the speed of light, too slow for distant stars zip past in a matter of seconds as shown. Some fans explains that these aren't stars but dust particles that interact with the warp field.
Space: 1999 comes through again by having characters track the approach of faster-than-light craft optically, and by allowing floating space rubble, conventional rockets, alien spacecraft, and a moon hurtling interstellar distances in days to be in range of each other for exactly as long as the plot demands.
In Babylon 5 hyperspace travel is, as per the DVD commentaries, done at the speed of plot. For example, it takes 3 days to get from Babylon 5 to Earth, which is about 14 light-years away. It also takes 3 days to get from Babylon 5 to Z'ha'dum, which is about 20,000 light-years away. It's explained that Hyperspace travel doesn't necessarily translate linearly. Which might explain why no one on the show ever says how far away something is, only how long it will take to get there.
Stargate SG-1 and Stargate Atlantiszig zag this trope. One the one hand, the Daedalus is able to reach Atlantis for the first season finale in about three days while powered by a ZPM. The Pegasus Galaxy (assuming the one in the series is the Pegasus Irregular Dwarf Galaxy in Real Life as is implied) is 3 million lightyears from the Milky Way. That's a speed 365 million times the speed of light. It's also mentioned that the trip would take about 2 weeks without the ZPM. That's about 78 million times the speed of light. Taken by themselves, these don't seem horrifically unreasonable. The universe runs on MORE POWER!. The problem comes when one considers that it usually takes them a few days to get any where in the galaxy. At the aforementioned speeds, they could traverse the entire observable universe in and 55 and 255 years, respectively. So either they normally run their engines far below their capacity, or it's this trope.
At one point it was explicitly stated that ships had separate drives for interstellar and intergalactic travel, for whatever reason.
Handwaved in Warhammer 40,000, where the chaotic nature of the Warp means that the same trip can take vastly different amounts of time. We have at least one example of a ship arriving quite some time before it left, and another of a vessel arriving at its destination millennia after it was supposed to arrive, though the crew seemed to believe it took only a couple of months. When your hyperspace is made of illogical thought, it's no wonder there's a bit of variation in travel times.
In one case this is used for some dark humor. A crew of a ship responding to a distress signal travels through the warp ending up being attacked by demons. They proceed to send out a distress signal... which is sent back in time due to the anomalies in the warp, the very signal that they themselves responded to.
40K also came up with a nice solution for the Necrons: the Inertialess Drive. It makes the ship unbound by inertia, allowing it to almost instantly (the time it takes for the drive to activate) accelerate to a practically infinite velocity, then immediately come to a stop. Necron phase technology would prevent them from crashing into anything, and the precision of their machinery (their weapons, for example, are crafted atom by atom, as an idea of how precise Necrons are) would ensure they don't go waaaay too far, out past the galaxy.
It was actually E.E. Doc Smith who came up with this idea for the classic Lensman series.
And retconned away in the 5th edition Necron codex: the Necrons use the same wormhole network as the Eldar by forcing their way in. Also, being immortal they can and do simply sail between systems at near-light speed. Slow, but infinitely more reliable.
Nova Praxis claims that rail guns and coil guns in the setting fire projectiles at 18000 feet per second. This is almost 5500 meters— per second. Even assuming that a bazooka-sized gun shooting a BB-sized projectile at such a speed would put the recoil at near 7500 Newtons; plenty enough to crush any bones in the way if not completely dismember your arm. Ironically, that's the least of your worries since friction from the air would heat your BB up to five times the surface temperature of the sun, forming a quickly mushrooming cloud of burning plasma in its wake. The person shooting such a gun would surely be killed in a very quick and horrible manner. The target has a much better chance of surviving as the shooter might have missed.
In System Shock 2, it's discovered that a piece of the space station which was jettisoned by the player in the first game has crashed on a planet in the Tau Ceti system - crossing a distance of 12 light-years in a mere 30 years. This would require the ejection charges to have kicked the module loose at about half the speed of light...
Then there's the issue about crashing the planet at half the speed of light...
Ratchet & Clank Future: A Crack In Time has the titular heroes caught by Dr. Nefarious and propelled off his space station on a catapult-like device to their assumed deaths. They awaken on a planet "hours later". A large catapult is not likely to propel anyone to fast enough speeds to reach a planet outside presumed detection range without turning them into paste. Also may qualify as a Distance example.
When comparing the distances and sizes of moons and planets in Crack in Time, this makes considerable sense. In R&C, the planets and moons really are that close together.
The X-Universe series has a particularly egregious example in which each individual map is at most two hundred kilometers across, and as little as tens of kilometers in the original Beyond the Frontier — almost comically small by astronomical standards — yet your ship requires a Time Dilation device to travel between locations on the same map in a reasonable amount of time. Unless the kilometer was redefined at some point, this suggests spaceships in the game are far, far slower than they have any right to be — raising the interesting question of how any of the spacefaring races actually managed to become spacefaring races when they don't seem to have any ships that come anywhere near escape velocity for a planet with a mass similar to Earth. For added fun, the ships we actually see entering and exiting planetary atmospheres in the series are TL-class large transports, of which the absolute fastest in the series, tuned for maximum speed, can go about 1,000 km∕h, and which are more typically in the 200-400 km∕h range.
Star Ruler: You can travel faster than light on raw engine power alone, no Phlebotinum handwave provided. Lasers also remain hitscan even as their effective range becomes measurable in AU, which means that they are travelling FTL as well. However, the rest of the game generally is relatively "hard", but can go into Misaimed Realism when one has high-tech ships such as the aforementioned FTL lasers.
In Evochron Mercenary, it's possible to see a galaxy. The galaxy is rotating at about the same visible speed as a carousel, meaning that it is rotating at several hundred billion times the speed of light. Ships in the game are capped at 10000 meters per second in the X, Y, and Z plane.
A Mass Effect example caused by a minor continuity error: In the War Assets for Mass Effect 3, the Turian Spec Ops Team's entry mentions they came from a turian colony world named Taetrus where their home city of Vallum was destroyed by a terrorist ship ramming it at FTL speeds. The energy release from a ship impacting at such speeds would do tremendous damage, possibly destroying the entire planet depending on velocity. However, the larger story surrounding Taetrus' civil war and the Vallum blast was detailed in the Cerberus Daily News series (a collection of daily news snippets released between Mass Effect 2 and Mass Effect 3) which specified that the ship was not traveling at faster than light speeds and in fact only impacted at the speed of a single mass accelerator round.
Although, the fact that FTL travel in-universe is accomplished by reducing the effective mass of the object in question (hence the title) means that an object moving faster than the speed of light would impart a significantly lower(read: possible) amount of kinetic energy. Likely still higher than the amount the fully massed ship would have imparted, moving slower than light, but that depends on how much the mass was reduced in the first place, as well as at what point the mass effect generator inevitably failed during the impact(when the electrical current stops, the eezo stops generating the mass effect field) as well as whether the ship's mass was negative, or simply proportionally reduced at the time (or, indeed, if negative mass is possible, in-universe; the codex description of FTL travel is rather limited).
In The Magic School Bus episode "Out of this World", the class has to stop an asteroid from hitting the Earth, specifically their school. It takes the whole episode for the asteroid to get close to Earth and that's with Dorothy Ann having tracked it for days prior to the start of the episode. They defeat it by changing its trajectory to Hurl It into the Sun, which it manages it hit within the span of a few seconds. This is Lampshaded in the producer segment ("Our show is less than thirty minutes long, what could we do?")
Although this doesn't involve space ships, the roleplaying site Mega Man MUSH once had a memorable example of this in its news files for the various character stats, describing what the specific numbers for each stat would represent. In the news file for the Velocity stat, where 1 signified "less than 5 mph (8 km∕h)", 5 meant "60-150 mph (97-241 km∕h)", and 9 was specified as "767 mph (1235 km∕h)" (the speed of sound), 10 was defined as "escape velocity". Hilarity ensued when someone pointed out exactly how fast escape velocity is.
To those who aren't into reading through the math in that link, Escape Velocity is 11.2 km∕s, or over 40,000 km∕h, thus leaving a drastically large gap between ratings 9 and 10.