Turnbull I agree with you, its a very hard question and it takes ALOT of discipline not to give a summary of each movie, with my favorite scenes, favorite dialogue, plot, but suffice it to say I can watch these below, over and over.
Also note: Those folks who have been asking about clothing and fedoras... Cagney can surely dress in some well tailored suits and hats!!!

Angels With Dirty Faces (a great plot, sad AND Funny)
White Heat
Public Enemy
Roaring 20's
Each Dawn I Die
The Fighting 69th--(War movie but good)
Yankee Doodle Dandy (Cagney can DANCE)
African Queen (A great 'non gangster' film)
Treasure of the Sierra Madre
Casablanca
High Sierra
Key Largo

If you like this forum you'll love the movies above. You MUST buy 'em, tape 'em, see 'em and enjoy 'em by any means necessary. #@*&*! I'm long winded. Sorry.
Some are definitely on vhs and they'll probably come out w/ dvd's.