These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Adaptation Displacement: Sort of. Casablanca was adapted from a stage play called Everybody Comes to Rick's, which took place entirely in the cafe and ended with Rick and Ilsa running away together, though few people know this. They could be forgiven for not knowing, however, as the play was never actually produced.
Alternative Character Interpretation: The classic interpretation is that Ilsa is truly helpless to her feelings for Rick, and his words to Victor are a noble gesture to put his romantic rival's mind at ease and make sure they'll be happy together. However, Ilsa tries everything from reason to begging to threatening Rick with death in order to get those papers for Victor before she kisses him and finally melts his reserve. And remember, this woman has been sticking with Victor through some truly hairy situations, the willing and loyal accomplice to the most wanted spy in Nazi-occupied territories. Whatever feelings she may still have for Rick, it's no stretch to imagine that she's faking that helpless damsel act, and willing stay with Rick if it that's her only means of getting her husband to safety once and for all.
More evidence: Ferrari offers them one letter of transit, and she never says, "Honey, you're the one they're after; take this one and get out to continue your work." She doesn't float the idea of staying behind until she's tried everything else she can think of. Also, she hears a singer with a familiar voice, is told that his name is Sam and that he came there from Paris with Rick, and she's still asking who Rick is. Doesn't exactly sound like she's been pining for him.
Or it might just be Conrad Veidtbeing Conrad Veidt. He was a fine actor, bless his heart, but watch any of his sound films and his flamboyant bisexuality always leaks through. (Note that Strasser does not only leer at the guys, but also leers at Ilsa...)
Also—and let's be honest about this—anything Nazi is going to be at least a little bit gay-seeming (as Jon Stewart & Co. put it, Hitler basically turned all Germany into "a smelly, homoerotic gym class").
Awesome Music: When Victor Laszlo leads the crowd in Rick's in La Marseillaise to drown out the Nazis' "Die Wacht am Rhein". Doubles as a Moment Of Awesome for Victor.
Start humming it people. "You must remember this / A kiss is still a kiss / A sigh is just a sigh..."
Director Displacement: Michael Curtiz won the Academy Award for Best Director, but that's about all the acclaim he's ever gotten for this movie. Film historians and critics usually credit the producer, Hal B. Wallis, and the various writers (Julius and Philip Epstein, Howard Koch, and the uncredited Casey Robinson) for the creative direction of the film. At worst, Curtiz has been called a "journeyman hack" or a "hired gun". His overall career was a success, however: He directed 173 films, starting in 1912, and didn't miss a year until his death in the early 1960s. Many are unremarkable, but there are at least a dozen that are still well-regarded.
Fridge Brilliance: Ugarte fires four shots at the police while trying to escape. Most guns contain six bullets. He already used two bullets on the two couriers he murdered.
Genius Bonus: Bogart's eyes are brown - not that viewers could tell that, since Bogart had appeared only black and white films until that point.
Harsher in Hindsight: Victor's defiant line "Even Nazis can't kill that fast." This was written before the general public knew at the time of the millions of Jews in Nazi concentration camps being killed at an unprecedented speed via methods like gas chambers.
Similarly, Peter Lorre's Ugarte screaming for Rick to "Hide me!" as he is about to be apprehended (and killed) by the Nazis. Lorre was a Jewish actor born in Austria, who worked in Germany until the Nazis came to power. He ended up departing for France and later to America.
Marcio Dalio plays the roulette table minder. His most famous role outside of Casablanca was as an ineffectual French noble in The Rules Of The Game. Combined with the repeated hints that most of Rick's staff have fallen very far from their formal social station, it can be a little discomforting.
Honorable Mention must also go to Ugarte, who really loves to invade Rick's personal space. Of course, this is just part of Peter Lorre's trademark style of acting. Ugarte may not come across as exactly "gay" or "bisexual" so much as simply an eccentric who will half-consciously make love to anything that gets too close. (His cigarette got the most action.)
Strasser seems to be checking Renault out several times during the movie. Right after their conversation about bumbling Americans and Germans, Strasser says "As for Laszlo, we want him watched twenty-four hours a day" and blatantly eyes Renault's crotch and licks his lips before taking a puff off his cigarette. Hard to say whether that's Strasser checking Renault out or Veidt checking Rains out, though.
It Was His Sled: The plane scene and "We'll always have Paris" is... uh... kinda obvious. It's even the iconic scene shown at Disney's Hollywood Studios for its famous movies tour.
Memetic Mutation: A good chunk of lines from the film have become considerably memorable. Notable is "Play it again, Sam", but they don't say these words exactly.
"This looks like the beginning of a beautiful friendship."
"I am shocked, SHOCKED to find that (insert not-so-shocking event)."
Sequel: After a fashion; it's the novel As Time Goes By, by Michael Walsh. It picks up exactly where the movie left off, sends Rick & co. off on a mission for British Intelligence, and delves more deeply into Rick's backstory. A clear-cut case of First Installment Wins.
Tear Jerker: The "La Marseillaise" scene, especially Yvonne's close-up which was possible the actress's real emotions. The movie was shot in 1941, when France was still occupied by Nazi Germany, and the movie was cast with many refugees with memories of their home being invaded fresh in their minds. It would be similar to a movie about 9/11 shot only months after it happened with some of the actors having lost family in the attacks.
Rick having to remind Sascha to escort the drunk woman home and then come right back. Sascha's disappointed "Yes boss" implies he had been hoping to take advantage of the situation.
Similarly, Ilsa referring to the adult Sam as a "boy."
Along with a touch of Fair for Its Day — Rick clearly respects Sam, and it's heavily implied that Sam stays with Rick out of genuine concern for Rick rather than money or foolish loyalty.
Watch It for the Meme: Watch it for any of the most famous lines; possibly even watch it for the Beam Me Up, Scotty!, as many people now realize that it isn't "Play it again, Sam." In fact, the majority of this movie's most famous lines are in the end scene, causing it to practically overdose on Memetic Mutation in the last ten minutes or so.
Xanatos Speed Chess: Possibly Ferrari, behind the scenes. We know he wants Rick's Cafe and all of the talent Rick has running it. So, when Ilsa and Victor approach him, he offers: "I am moved to make one more suggestion; why, I do not know, because it cannot possibly profit me, but, have you heard about Signor Ugarte and the letters of transit...?" In the end, he gains Rick's Cafe and most of the talent running it, so it did profit him. He also has little to gain from the Nazis closely watching the black market.