Fleas do jump. They have been reported to jump up to 7 inches vertically and 13 inches horizontally, making them some of the best jumpers on the planet relative to body size.
Researchers have discovered that fleas do not jump from their trochantera, which are the insect equivalents of knees, but rather from their tibiae and tarsi, which are the insect equivalents of feet. They do not jump with direct muscle power but store energy in the form of a protein called resilin. When the flea wishes to jump, it releases the stored resilin rapidly. This acts similarly to how a human uses a bow and arrow, and the release of energy causes the flea to jump.