Can you feel passing through the sound barrier in an F-16?
In an aircraft designed to go supersonic, it's an absolute non-event, and one is only aware of it by observing the instruments, and noting diminished control authority-- slower roll rate, etc. At least, that was my experience in the T-38, and according to every account I've read. 

If the aircraft is NOT designed to go supersonic, then the experience can be quite different, although that mostly comes from loss of control at high Mach numbers (starting in the 0.9X range, as best I recall), rather than anything that occurs right at Mach 1.00.