I would say that 'stutter' describes when the flow of speech is interrupted, usually associated with speech problems, as with 'stammer', if somebody has a stammer, they may repeat a word or hesitate with words during sentences. Whilst 'stumble' is generally associated with something physical, like a fall - somebody might accidentally stumble/ or lose their balance, or you could 'stumble upon' something, which would mean to find information, which isn't falling over at all!