In "Old Iron", James runs away out of control after some boys fiddle with his controls. Edward manages to catch up to him, and his fireman brings him under control.
In "Toad Stands By", after Oliver's accident, the trucks on Sodor tease him constendly, singing songs and playing tricks. They're lead by S.C.Ruffey, an old privately-owned wagon. What does Oliver do? He pulls S.C.Ruffey apart. The trucks have never played a trick on Oliver since.
As pointed out by GeekGo of Channel Awesome, it was also the first time a truck had been DEFEATED, instead of injured or punished.
In "Gallant Old Engine", in a flashback, Rheneas has a moment where after suffering a jammed valve gear on one side, he still struggles onward with his train.
Made even better by the Awesome Music which plays while he struggles on. And he HAD TO finish his journey, or else the railway would close.
In the episode "Four Little Engines", Skarloey continues on even though badly worn out and now with a front spring broken.
In "Edward's Exploit", when Edward gets the train rolling again despite his broken crankpin and that triumphant reprise of his theme starts playing. The passengers cheer when he arrives at the station, even though he's late.
In "Bowled Out", a snooty diesel fails after sucking a bowler hat through his air intake, when due to take an express (15 coaches in the original book). The only engines left are Duck (an ex-GWR 57xx tank engine, built in the late 1920s for light passenger, station pilot duties, and shunting), and Stepney (an ex-LBSCR A1X "Terrier" tank engine, built in the 1870s for even lighter passenger, station pilot duties, and shunting). They manage to get the train to their destination early. And to cap it all off, this is the last duty Stepney performs before going back to his slightly less hectic preserved line in England.
In "Percy's Promise", Percy manages to bring a Sunday School excursion train back to the end of Thomas' Branch Line through a flooded section of track. And the TV episode featured the Storm Theme, which would later be played in "Gallant Old Engine".
Duke holding back Sir Handel after the latter slips off the rails and almost falls off a cliff.
Duck saving an entire train of passengers from runaway freight cars in "A Close Shave". The dialogue afterwards says it all; "I didn't know you were a brave engine." "That's alright, sir, I didn't either."
Edward arranging for Trevor to be purchased by the Vicar in order to save him from being scrapped, and, inspired by this CMOA, Douglas (and his drivers) saving Toad and Oliver.
And then Rusty, inspired by Douglas, saves Stepney.
Duck standing up to the big engines in "Duck Takes Charge". Seriously, until he showed up poor Percy was picked on by them endlessly. It's not hard to see why Duck and Percy are good friends; Duck was the first engine who stood up for the little guy!
Jack jumping in and saving Thomas from falling off a bridge in "A Friend in Need".
Byron saving Alfie from sinking into a mud put in "A Happy Day for Percy".
Alfie saving kittens from being crushed by a collapsing building in "Alfie Has Kittens".
"Rusty and the Boulder"
"Thomas and Bertie" race.
Terrance to the rescue in "Thomas, Terrance and Snow".
The Fat Controller saving Stepney in "Stepney Gets Lost".
Stafford gets one in his debut episode. After being bossed around by Spencer all day, Spencer gets himself stranded without any coal. Stafford nonchalantly shunts him all the way to Knapford Station.
Thomas' very first CMOA, saving James in "Thomas And The Breakdown Train"/"Thomas Saves The Day". Extra points for making the troublesome trucks, who made a fool of him in the previous story, whimper and cower before him.
Thomas saves James from a fallen tree five seasons later in "James and The Trouble With Trees".
Mavis pulling Toby to safety off a decaying bridge in "Toby's Tightrope".
Mirrored in "Thomas And The New Engine" when Thomas saves Neville, Annie and Clarabel from toppling off the ledge of an unfinished bridge.
Pretty much anytime Gordon or James don't get a Break the Haughty moment and manage to prove what splendid engines they really are. "Down The Mine" and "Gordon And Spencer" (for Gordon) and "Troublesome Trucks" and "James and the Express" (for James) are good examples.
Gordon actually proved capable of bringing other engines down to size as well. He swiftly proved annoying prankster engines such as Thomas and Percy to be out of their league ("Thomas and Gordon"/"Thomas Gets Tricked", "Percy and the Signal") and can sometimes put slacker engines in their place in a very blunt, nonchalant fashion ("Scruff's Makeover").
Gordon: You may be shiny and clean, but if you don't do your job, then you're not a really useful engine any more.
In "Trouble In The Shed", the tender engines tire of shunting and go on strike, with Edward getting the scab treatment and insulted and hissed at constantly. The Fat Controller then hires Percy, who befriends Edward. Later, as Henry comes by hissing again, Percy outdoes him with a loud, enormous wheesh, sending Henry rushing back into the shed startled.
In "Edward, the Really Useful Engine", Edward pushes behind both Gordon's Express and Duck's goods train up Gordon's Hill. In short, he was the back engine for two trains. This is intriguing considering that Gordon earlier thinks Edward should retire in not a moment too soon and also claimed that he doesn't need a back engine.
"Henry's Hero" is filled with this. Henry and Hiro both end up taking low-quality coal, which ends up clogging their works. Henry backs out of his jobs out of caution, leaving Hiro to take the train they were currently taking. He finds the long line of stone hoppers too heavy to take on his own with the bad coal, so what does he do? He heads for a siding, and then takes half of the train to the docks before returning for the rest, even as his boiler becomes more and more clogged with ash. After being assigned a steel-girder train, he manages to persuade Henry to take it with him, but then breaks down. This leads to the biggest CMOA of all in the episode. Henry, who has regained his confidence, almost effortlessly manages to push the entire train along with Hiro up Gordon's Hill and the entire rest of the way. After suffering from many seasons of Badass Decay, both Henry and Hiro manage to prove their worth as heavy-goods engine for the railway.
"Not Now Charlie" is one for Charlie, whom many older fans find annoying due to his jokes. He finds an elephant on the line and actually takes responsibility for once, trying to get help to get the elephant to safety. Even though his friends think it's a joke, he still takes action and heads over to the Rescue Center to get help from Harold, who helps track down the elephant. After four seasons of being an irresponsible idiot, Charlie finally takes action and is rightfully praised for preventing a dangerous situation.
Percy saving Bill and Ben from toppling over a hill in "Percy's Lucky Day". As something of a meta example, it's somewhat cathartic to see Percyfinally having a good day where everything goes swiftly (perhaps most notably when he is given a special and, for arguably the first time for any engine since the CGI transition, delivers it effortlessly with nothing bad happening whatsoever).
While sadly an Offscreen Moment of Awesome, Thomas somehow saving the Fat Controller and his runaway sleigh in "Santa's Little Engine".
After being absent for 5 Seasons and not having a starring role since Season 5, Toad the Brakevan gets his when he manages to stop James' runaway train (caused by a fallen branch) from crashing into Thomas and Percy. He even sticks up for James afterwards when the Fat Controller reprimands James for his carelessness.
James owning up to his mistake is arguably this as well.
Two in "Thomas Saves the Day" (the Season 8 episode), Thomas making it through a sharp bend without Annie and Clarabel's help. Right afterwards, he manages to alerts the signalman to divert a set of runaway trucks from crashing into a station, wrapping up both plots nicely. On a meta-note, it's also considered to be the highlight of an otherwise subnormal season.
On a meta level, Andrew Brenner becoming head writer is awesome. He has brought new writers in and dramatically boosted the maturity of show. At the same time, he was brought back old characters like Duck, Oliver, Bill and Ben much to the fans delight, as well as toning down Thomas' Spotlight-Stealing Squad tendencies. The whole fanbase agrees that Andrew Brenner has caused Thomas the Tank Engine to Re-Grow Its Beard after Sharon Miller's Seasonal Rot.
In Thomas and the Magic Railroad, Lady being steamed up for the first time in decades, and puffing away down the titular Magic Railroad. It mingles with Crowning Moment of Heartwarming.
The chase scene, especially the climax when the viaduct collapses in front of Thomas, and he doesn't even slow down. He just goes for it.
Even more impressive in the '99 script, where Lady is actually turned off onto a siding right before the bridge, and Thomas, aware that it's collapsing, deliberately lures Diesel 10 and Boomer onto it, declaring, "I'm not afraid! I'm not afraid!"
Another awesome scene is Toby delaying Diesel 10's plot by ringing his bell and causeing Pinchy to knock over a shed in the confusion, trapping the diesels.
Thomas vs. Spencer in the climax of "Hero of the Rails".
Gordon gets a Big Damn Heroes moment, after Spencer had chased Thomas and Hiro, leaving Hiro broken down in the middle of the line. He shows up and spontaneously challenges Spencer to a race, knowing he can't resist. Not only did he end up winning the race, but also got Spencer in trouble with his owners!
Thomas pulling Diesel to safety after nearly toppling over the unfinished bridge.
In "Day of the Diesels", Percy rallying the entire Steam Team together to have them take back the Steamworks from the diesels.
Flynn and Belle stopping the Dieselworks from burning down.
Diesel 10 nonchalantly crushing a peace of junk with Pinchy, making Diesel 10 look Badass.
Thomas pulls off a big "Screw the Rules" moment in "Blue Mountain Mystery": He goes on to Narrow Gauge tracks, putting himself at risk, just to tell Luke that he is innocent of the crime he believed he committed.
And doubled again just afterwards, when Luke, despite having refused to talk to Thomas just moments prior, immediately runs out and saves "his friend" from falling off the tracks.
In King of the Railway, Thomas rescues Stephen from an abandoned mine shaft. As they're leaving, the bridge beneath them starts to break apart. Jack jumps in and holds the bridge up long enough for Thomas to pull Stephen to safety.