Awesome Music: A lot of it. Particularly worth mentioning, in the film adaptation the overture includes a long, virtuosic violin cadenza performed by Isaac Stern.
"Sunrise, Sunset" is still performed at weddings to this day. And now there's two same-sex versions written by the original lyricist, Sheldon Harnick.
Broken Base: Topol being cast in the film instead of the original Tevye Zero Mostel was extremely controversial, with some fans even accusing it of sending the fragile-egoed Mostel into a depression that ultimately led to his death six years later.
Ear Worm: "Tradition" (especially the fiddle solo), "L'Chaim", "If I Were a Rich Man", and "Matchmaker, Matchmaker" are probably the worst.
Fridge Horror: At the end of the play there's a sad undercurrent, because Hodel and Perchik will apparently remain in deepest Siberia instead of escaping to Krakow like Chava and Fyedka. Then you remember that this takes place about 35 years before the Holocaust, and realize that Chava and her children will likely end up gassed in Auschwitz while Hodel and Perchik may very well survive.
Siberia was not left untouched by persecution either: in the Civil War, it would be fought over by both the Red Army under Trotsky (who, contrary to popular belief, would casually endorse a pogrom or two once in a while to help "purify" the "counterrevolutionary influences"), and the White army under Kolchak (who, if anything, was even worse in that regard). While it is quite possible that they would escape notice by not being near the railroad, they aren't likely to enjoy themselves either.
Of course, being a revolutionary exiled to Siberia, there is a chance that Perchik and Hodel are released following the Russian Revolution and Perchik eagerly joins the newly formed NKVD... making his storyline in the play a Start of Darkness.
Still, thirty years is plenty of time for the others to end up following Tzeitel's parents to America at some point (and hopefully Chava and Fyedka, who go to Krakow, would make the same decision, as Chava and their children would still be considered Jewish under the Nuremberg Laws). Poland had issues with anti-Semitism even before the Nazi invasion, and so perhaps they would have found another reason to pack up and leave. For that matter, they would have been eligible under the Right of Return.
And oh yeah, just to complete the picure: if Anatevka existed, Anatevka's confirmed to be in Ukraine, and there's a good chance it's within the radius of territory rendered uninhabitable by Chernobyl. Holy cow.
...You know, let's just draw the line here, and agree that any way you cut it, someone's gonna lose a finger. Fortunately, in the original stories, they pack up and move to Palestine, where they will have peace and accept-ah, crap!
"Funny Aneurysm" Moment: Not exactly funny, per se, but Perchik participating in the October Revolution definitely counts.
Germans Love David Hasselhoff: The play was a surprise hit in Japan, where the theme of crumbling tradition resonated heavily with the elderly generation.
Nightmare Fuel: The movie version of Fruma Sarah's solo in "The Dream".
One-Scene Wonder: Fruma Sarah. It's also a theatre role where hamming it up is required, along with a harness (or a cast member with a sturdy set of shoulders).
Protagonist Title Fallacy: Teyve is not the fiddler on the roof; however, the fiddler represents the inhabitants of Anatevka: trying to play a pleasant old tune in perilous circumstances.
Values Dissonance: The original Tevye the Dairyman stories by Sholem Aleykhem, on which the musical is based, portrayed Tevye's decision to disown Chava after she marries Fyedka in a more positive light. It was what any good Jewish father of the time would do if his child chose to marry a Gentile (and to really hammer this home, Sholem Aleykhem has Chava abandon Fyedka and return to Judaism at the end). People today who see the musical (which doesn't really take a side), though, usually come away from it thinking Tevye's actions toward her are excessive and cruel.
Though the movie does show that the father feels trapped by his traditions at this point, but also that she was kinda stupid to run off and expect everything to be hunky dory. Still, it is easy to see it as a Kick the Dog moment for Tevye.
And it's important to remember that it's not just culture, it's religion. By accepting Chava's marriage he would be rejecting his faith, and he wasn't willing to do that. In fact, to him his daughter rejecting their faith to marry a Christian is worse than her dying, as he believes it has eternal consequences.
He does unbend a little bit at the very end when the family is having to leave their land and Chava comes to say goodbye. Specifically, he gives her a quiet "God be with you" and subtly endorses his wife and other daughter's reaching out to her.