At the end of 'Coliseum', Hakeev of the Tal Shiar corners you. You've been cut off and betrayed, and you're staring down the barrels of a dozen Romulan disruptors. He happily gloats about your impending demise, his men level their guns and squeeze the triggers... and your ship locks onto your signal and beams you out just in time. Cue an outraged, perfectly-timed, and utterly pricelessBig "NO!".
Mission completion texts for Duty Officer assignments can be very amusing - especially when things go wrong.
Authorise Shipboard Concert - Failure:
In the wake of the concert, members of your crew have requested an unusually high number of amnesia-inducing stimulants.
Host Diplomatic Reception - Disaster:
Your victims were routed to Sickbay with alcohol poisoning.
Capture and Deliver Decendants of Caitian Unforgiven to Ferasans - Failure:
By the time the target was located, the Caitian had died of natural causes.
Or someone gets injuried and/or killed on completly mudane tasks. There have been reports of crew members having to be hospitlized after attempting to review sensor logs. The joke was that they got into a fist fight.
Dialogue Option 1: "Huh, you're right, I never thought about it that way."
Dialogue Option 2: "I'm glad to know someone agrees with me."
"Got any good gossip?"
[Morn rattles the drink in his hand.]
The best one is if you ask where he's been this whole time.
Morn ponders for a moment, gathering his thoughts on his grand adventure ... [ Gilligan Cut ] Several hours later... Morn finishes his tale. He looks at his glass and takes a long drink, his throat parched from the long tale. He nods and bids you farewell.
This one is very meta, but dammit, how Denise Crosby met Al Rivera (Capt Geko) and Christine Thompson (Kestral) of Cryptic Studios after agreeing to the job is nothing short of Crazy Awesome and funny as hell. Watch the whole thing here: 
First, this was a fan convention, the devs just happened to be going and learned there was an open Q&A
Al surprised Denise by revealing who he was to her who he was. He was the calm fan boy. SHE WAS FREAKING OUT.
Al/ Captain Geko : "My name is Al Rivera and I'm the leader content designer of Star Trek Online."
Denise Crosby: "THIS IS THE THE MAN I'VE BEEN LOOKING FOR!!!"
hevach: Clearly Cryptic needs more strict rules regarding alcohol in the office. Borticus (A dev): Quiet, you! >:|
Another one: "Sela still doesn't have a cape. - Now the cape makers are on strike. "
Yet more dev snarkiness. The servers died for several hours on March 21, 2014. The @StarTrekOnlineTwitter feed gave the following status report:
"@YouriAshton We're still looking at ridding ourselves of the Borg-pests. :)"
In the first of the new Klingon Empire 1-20 missions after the Tutorial, you end up fighting Naussican pirates in the Warehouse at the the Exchange. All throughout the fight, the Ferengi in charge of the Exchange is running around screaming in fright, but you can shut him up by punching him.
Just the fact that you can breed a Cannibal tribble that then eats all the common tribbles.
Heck, just the fact that you can equip a Tribble.
The Risa Summer Event. Starfleet and KDF Command find it so important, they call a temporary truce so that both sides can go to Risa, search for Horga'hns (the little statues that help you get laid by a Risian) fly jetpacks, play dabo and have a dance party. ... You guys remember you're at war right?
You can even find an expy couple of Worf and Jadzia fighting at night and being told as a federation captain that you cant learn Mokbara on the rocks by a Klingon captain.
As a bit of a bonus for the countdown to Season 8, Q has returned to the faction bases to grant you extra XP. One of the options you can choose is "Are you really as powerful as they say?" He promptly turns you into random creatures, up to and including and Undine!
It's even funnier when, if you're a Vice Admiral, you pass by the class as any of those forms and they salute you as usual!
Kestral's companion story to Sphere of Influence, "Of Chaos and Kal-Toh" was recently published in Star Trek Magazine and its pretty much Comedic Sociopathy:
Shon is being diessected by the Solanae while this is all going on
The Enterprise-F, ordered by Starfleet to investigate the literally just appeared out of thin air Jouret Gate, gets scanned by an Iconian Probe and starts falling towards the Sphere's surface ala Star Trek: Into Darkness.
Commander Winters ends up having to be a Turbolift Operator
Lt. Kyona is forced to enter the Jeffries Tubes (Which she hates)
Lt.Cmdr Tem Inasi is cheerfully informing the Bridge Crew how horribly their gonna die. ("The last 30 seconds should be very uncomfortable!")
In the "Party Crashers" event for the game's anniversaries, you have to chase down Tiny Qs to get them back up to the big (normal) Q. At one point, a Q can transform into various forms to run away. There's a chance that one of those forms will be a Tribble.