Here are some classic B&w films from my top list in no particular order.
Wages of Fear - desperate men transporting dynamite across dangnerous roads
Rififi - a 3 act movie on a jewel robbery. Act 1 -the plan and assemble the gang, act 2: a 30 minute wordless and near silent section of the robbery proper. And act 3 - of course, it then all goes wrong. If this sounds familiar, remember every robbery film you've seen is a remake of Rififi - the first and the best !
Cassablanca of course, but also To Have and Have Not - which is pretty much the same plot - and Bogart acting alongside a young Lauren Bacall. Contains some of the sexiest scenes ever committed to film. Bogart and Bacall fell in love making the film and it shows. "you know how to whistle don't you?"
The Third Man - cinematic perfection in post war Vienna.
High Noon - classic western
To kill a mockingbird. In the colour era but a black and white classic all the same. And the much later Last Picture Show from the late 70's .
Then there's the Ealing Greats: Whiskey Galore, Kind Hearts and Coronets and much else.
What about Mel Brooks' b&w masterpieces - Young Frankenstein and High Anxiety.
Ice Cold in Alex already mentioned - a movie set in wartime but not a war film as such.
and the master or Samurai movies Kurosawa - Seven Samurai (remade as Magnificent Seven), Youjimbo (remade as Fist Full of Dollars, and again as Last Man Standing), Sanjuro, Roahomon. All masterpieces - even before his colour work.
Scifi - there's Metropolis, and Things to Come.
I could go on and on
