Award Snub: One of the most famous in the history of the Academy. The film lost the Best Picture Academy Award to Shakespeare in Love due to the Weinsteins heavily lobbying their movie. Saving Private Ryan is the film people remember more these days.
Broken Base: Upham's actions while Mellish is being killed. It sparks numerous debates among fans. Mostly because it becomes an argument of "What would YOU do?" It's easy to say what you would do in that situation, and the internet (and real life, for that matter) are chock full of people who will disagree with you, whatever your claim may be.
Do Not Do This Cool Thing: As an extremely bloody and realistic war movie the natural expectation is that it would work solely as an anti-war movie by showing the horror. However, there is a non-trivial section of the audience that found the (lavishly shot) action scenes exhilarating, even glamorous. These reactions are far from mutually exclusive.
Of course, your opinion may take a 180 degree turn when the medics try to stop a soldier from bleeding out, or during the Telegram scene.
Ensemble Darkhorse: Jackson and Wade seem to be the favourites among the fandom. Their deaths seem to elicit the most Tear Jerker reactions anyway.
Fanon: Many people who saw the film were confused as to whether the German soldier that kills Corporal Mellish is also "Steamboat Willie." They are in fact, different soldiers. The soldier that kills Mellish has Waffen SS lapel insignia, while "Steamboat Willie" has the lapel insignia of an enlisted soldier in the Wehrmacht Heer.
Fridge Brilliance: At one point of the film Miller and Horvath discuss Ryan and their mission, with Miller stating that Ryan had better return home and cure cancer or invent the longer lasting light bulb. Later on he tells Ryan himself to "earn this". The elderly James Ryan tells Miller's grave site that he has tried to live a good life and he hopes that in Miller's eyes this was enough. Not exactly the ideas Miller himself mentioned, and there has been a lot of discussion on whether Ryan did earn what they did for him. However, in a sense, we are all Private Ryan. It is oftentimes said that the soldiers who fought and died in WWII did so for our right to freedom and the way of life we take for granted today. How many of us have cured cancer or invented the longer lasting light bulb? Does that mean we aren't deserving of the sacrifice those soldiers made? Have any of us truly earned it? It may seem that Ryan should have done more with his life but maybe then we all should?
Friendly Fandoms: With Band of Brothers, due to both being companion pieces by Spielberg (though he only produced the miniseries) and having very similar visual styles - especially during the battle scenes. Both projects had the cast endure a boot camp from Captain Dale Dye too.
Heartwarming in Hindsight: Private Ryan is a paratrooper in the 101st Airbourne. He refuses to go with Captain Miller because he does not want to desert his post and leave his comrades behind. Band of Brothers came out a couple of years later and focused on the real men of the 101st. After seeing truly how deep the men cared for each other makes Ryan's refusal to leave his post all the more poignant.
Hell Is That Noise: The approaching German Tiger as the American soldiers were getting ready.
Dale Dye has a cameo where he talks about how the 101st Airbourne has been scattered across Normandy. This is slightly hilarious as of Dye's role in Band of Brothers, which covers the exploits of Easy Company in the 101st. It makes one chuckle that Dye's character in this doesn't know he has a lookalike in Normandy.
This is the first time that America put together an extremely dangerous rescue mission to bring home Matt Damon. It would notbe the last.
It Was His Sled: A minor example about an hour into the film. The unit finds a Private James F Ryan and breaks the news to him...only to find out it's not the one they're looking for. Spielberg's intention was to cast an unknown as Private Ryan and while Matt Damon had just recently netted an Oscar nomination for Good Will Hunting he still wasn't exactly a star. Nowadays everyone knows Matt Damon is James Ryan, not to mention Nathan Fillion being a fairly recognisable name too. So while it's revealed Fillion's Ryan is not the guy pretty quickly, most new viewers are bound to catch on before it.
Jerkass Woobie: Mellish is a dick throughout the film but then you realise he's Jewish, and there's also an early scene where he's handed a Hitler youth knife. He tries to joke about it but bursts into tears. Reiben as well, since he only wants to desert the mission after being forced to watch two comrades die slowly and painfully.
Narm: Played with, if that's possible with this trope. The scene with the wrong Private Ryan, played by Nathan Fillion, can come across as accidentally hilarious since everyone knows that Matt Damon is the Ryan they're looking for. So with that foreknowledge the scenes can come across as funny in a Crosses the Line Twice sort of way. But at the end of the scene Fillion's Ryan is still worried about his brothers and realises he won't be going back home. His line "I wrote them a letter just yesterday" eliminates any humour the scene may have had.
Retroactive Recognition: A lot of the younger actors, especially Matt Damon and Vin Diesel, were not yet megastars at the time this movie came out. Matt Damon had won an Oscar (albeit for Best Screenplay in Good Will Hunting), but it may yet have been a flash in the pan. Spielberg is rightfully credited as helping jump-starting Diesel's career, writing the role specifically for him after seeing Diesel in his acclaimed short film.
Jeremy Davies (Cpl. Upham) is Daniel Faraday in Lost.
Paul Giamatti and Bryan Cranston also have small roles and went on to become acclaimed actors.
Seinfeld Is Unfunny: Both on the giving (it's not the first movie to use a documentary-like depiction of war or realistic carnage, but the first popular one) and taking ends (the influence on war movies or battle scenes in general, complaints about the patriotic/emotional tone of the post-Omaha Beach part of the movie).