"Stammer" and "stutter" are synonyms. As always, there might be U.S./U.K. differences. I have an idea that "stammer" is more common in the U.K. and "stutter" is more common in the U.S. but they are both commonly used in both places. They both refer to speech impediments--a break in the natural flow of words, or an unwanted, involuntary repetition of certain consonants. "Stumble" usually refers to walking or running. It means starting to fall, but recovering. "I stumbled on the uneven pavement." Figuratively it can refer to many things. "I got the first three questions right, but stumbled on the fourth."

Essentially, they all mean the same thing. However, stumble is mainly used in relation to tripping over, although you could consider someone stumbling when they talk like this..."I erm, I didn't, you know, I don't think...". A stutter is often considered repeating a word due to difficulty to get that word out for instance... "but but but but but....". A stammer on the other hand is usually difficulty with pronouncing words. In general terms though they all can convey the same thing!