Given that not all recordings have it, it seems likely that Thenardier referring to Cosette as "Colette" in the "Waltz of Treachery" was thrown in at some point, and is fairly common in productions of the musical. In the film, this ad-lib was turned into a Running Gag.
During one rehearsal for "On My Own", the actress was so nervous she wrapped her arms around herself. The directors liked this, and it has been a part of the musical ever since.
The exclaimed "Enjolras!" at the start of the ABC Cafe Scene was originally an ad-lib that is now fairly common. (Notably this the only time Enjolras is directly called by his name throughout the entire musical.)
As recounted in the book Making It On Broadway: Actor's Tales of Climbing to The Top, John Rubenstein, the original Pippin, had very bad bunions on his feet that made him reject every pair of shoes that the costume department ordered for him. In a fit of pique, he resolved to do the entire show barefoot one night. It was uncomfortable, but he did it. Bob Fosse ran in to see him at the end of the night, and Fosse said, "John, I loved what you did tonight. Keep it in. Barefoot! Gives you that innocence." To this day, the title role in Pippin, wherever you go, is done barefoot.
Spamalot: The scene where the French soldiers are taunting Arthur and his knights has become famous for ad-libs. One famous showing of Spamalot in New York City featured the French soldiers spitting out "I throw both my shoes at you!" in reference to the recent incident where an Iraqi journalist threw his shoes at President Bush. The actors were visibly smiling and trying not to laugh on stage.
Also, "I headbutt you!" after the infamous headbutting incident in the 2006 FIFA World Cup involving the French player, Zidane.
Another from Spamalot: The Knights Who Said Ni have also become famous for ad-libs, and one showing received a lot of attention when one of the knights blurted out "And I am the father of Sarah Palin's daughter's baby!" The actors barely managed to stay in character.
Another — the evening after the announcement that Obama was to receive a Nobel Prize, the lead Knight Who Says Ni said they were changing their name to... a whole lot of stuff having to do with the Nobel, ending with, "And can we send Dick Cheney to accept the prize? Hey, it worked on Roman Polanski!"
Yet another: In a Chicago show a short while after Rod Blagojevich had been impeached due charges of trying to sell Obama's former Senate seat, one Knight conducted a one-man auction for the Senate seat in question and the French Taunter included the line "We impeach your crooked governors!"
When doing a show in Austin Texas, the Knights changed their name into the song "The Eyes of Texas" although they claimed it was actually the new Homeland Security motto.
In light of the Seattle Seahawks' Super Bowl victory, the 2014 5th Avenue Theatre production included a 12th Knight.
"Diva's Lament" originally had the line "They're trying to replace me with Posh Spice", which following Britney Spears' Creator Breakdown in 2008 was changed to "I'm constantly replaced by Britney Spears", then in 2014 was replaced again with a reference to Miley Cyrus' VMA twerking performance.
On the opening night of the musical version of The Producers, Nathan Lane improvised the line "Who do you have to fuck to get a break in this town?" in the middle of the song "The King of Broadway." It's been that way ever since, including in the movie.
Another from The Producers: Tony Danza played Max Bialystock on Broadway for a stint, despite the fact that it is referenced in the script that Max is fat and Tony Danza is certainly not. When they got to the iconic line where Bloom screams "FAAAAT!", Danza would look confused and say "...what?" and Bloom replied "You were fat... once...." and proceeded to call him "You once-fat fatty fat walrus!!"
Another example from The Producers involving Nathan Lane, during the London production of the show, directly after the scene where Max and Leo have just taken Siegfried Oath and are trying to leave Franz's rooftop. Lane, playing Max, tried to open the door leading offstage, but a stagehand had mistakenly locked it before the scene began. When the door didn't open at first, Lane shouted "We're trapped! Trapped like rats with a crazy Nazi!" When another one of the stagehands rushed to unlock it, Lane calmed down and said, "Got it." You can still see part of this in the 2005 film version.
In one of the performances of the 2004 revival of Stephen Sondheim's Assassins, the actor who played Gerald Ford tripped while entering the stage and fell flat on his face. However, since Ford had a history of accidentally tripping and falling (most notoriously down the stairs of Air Force One), the cast ran with it and it became a fixture of the scene in question for most subsequent adaptations.
During one of the rehearsals for the final scene of The Phantom of the Opera, the wedding veil Sarah Brightman was wearing fell off as she left the Phantom for the last time. Michael Crawford, in the title role, picked up the veil and buried his face in it, sobbing "Christine!" the whole time. It was left in and still continues to be an important bit of business for the character twenty years later.
In Patience, there is a scene where Bunthorne's admirers are commiserating on their unrequited love for him. Gilbert scripted the following exchange, minus the final line, introduced by the actress playing Saphir but liked by Gilbert so much it is now part of the script:
Ella: The love of maidens is, to him, as interesting as the taxes! Saphir: Would that it were. He pays his taxes! Angela: And cherishes the receipts! Saphir: Happy receipts!
The Mikado has perhaps more of this than any other Gilbert and Sullivan play (e.g. "Modified rapture!") Gilbert himself encouraged producers to do the same: The "Knightsbridge" line invites the director to substitute a place known for scandal at the time instead of saying "Knightsbridge."
At this point, it's probably easier to list the productions of Gilbert and Sullivan musicals that don't throw in a few topical or local references. Ko-Ko's "little list" song (which describes various people the world could easily do without) is particularly susceptible to additions.
Monty Python performed during The Secret Policeman's Ball variety concert, putting on (of course) the Dead Parrot sketch. Halfway through, though, Michael Palin began to break up, unable to control himself. Although the audience had cottoned onto the problem and was already chuckling, John Cleese covered magnificently and sent the audience into hysterics with one, ad-libbed, shouted line: "THIS IS NO LAUGHING MATTER!"
In the premier performance of the opera Rigoletto, the singer performing the title role, though experienced, was so unsure of his performance (due to wearing a prosthetic hump; Rigoletto is a court jester) that he suffered severe stage fright. The director/author, realizing this, shoved him on stage. The audience thought his stumbling entrance was a scripted joke and loved it.
During the run of the 1st version of The Scarlet Pimpernel musical, Douglas Sills (playing the title character) was notorious for ad libbing. Many of his ad libs became part of the scripts for the 2nd, 3rd and 4th versions of the show (yeah, it had a complicated run).
A Very Potter Musical: Harry's line about what he'd do with an invisibility cloak ("I'd kick wiener dogs") was an ad-lib when Darren Criss forgot his line. Also, Lauren/Malfoy's line "And-you-have-to-be-my-slave-for-a-whole-day-starting-now!!" was an ad-lib, judging by Joe Moses and Joe Walker's reactions to the line, given that Moses usually kept relatively straight-faced (well, sort of), and he was laughing in the background.
The "Oh, Goyle Rules!" line was thrown in by the actor as well.
Possibly subverted in A Very Potter Sequel when Ron/Joey Richter can't get the Taylor Lautner poster off the wall and proclaims, "It must be stuck on there with magic!" then when entering another room, "Wow. It's like every room in Hogwarts have been remodeled." This was either legit, or it was scripted for a good laugh.
It was legitimate. Nick and Matt Lang, the creators of the show planned for that to happen because they wanted the audience to be thoroughly surprised by what Peter Pettigrew really was. They later admitted that it made no sense to try that hard to trick the audience.
When a prop from USLES 's production of Aladdin was missing from the final performance (the ring which housed the evil genie) a replacement had to be found at short notice. The villain ended up being the Genie of the Shoe, and Abanazer threatening characters with cries of "Shooooooooeeeee!" ended up being the show's funniest running gag. Children in the audience were even asking to smell the evil shoe.
The sex between Kate Monster and Princeton during "You Can Be As Loud As The Hell You Want (When You're Makin' Love)" in Avenue Q was originally supposed to be under a blanket, but apparently one day during rehearsal, John Tartaglia and Stephanie D'Abruzzo started playing around with the naked puppets to see what it would look like over the sheets. The choreographer liked it so much that Avenue Q is considered by many to be puppet porn even today.
This is the very nature of Improv acting. During a performance, all of the actors do this, taking whatever the others do and running with it, no matter what it is.
The script book also mentions in a footnote that there is a section of time set aside somewhere in the middle of the play for Adam Long (or rather, his replacement as of 2003) to go off on an improvised rant about, to quote the book, "whatever bug he has up his butt at the time".
The song "Before The Parade Passes By" in Hello, Dolly! was written in half an hour to replace a different song, "Penny in my Pocket", which wasn't doing well in rehearsals.
A Little Night Music: Desiree's spoken exclamations in "The Glamorous Life" ("Where the hell is the stage manager?" and "No hot water again!") were originally ad-libbed by Jean Simmons of the original London cast.
In any show that has a flub (be it dropped line, missed cue, or even an issue with the set) the actors must find a way to continue the show without giving away to the audience that all hell has broken loose.