You have to save them manually yourself and you can only save them in single player mode.
If you hit a great shot and get an automatic replay, look for the "SAVE REPLAY" checkbox in the bottom left of the screen (where your clubs normally are). Make sure to check it before the replay is over.
If you don't get an automatic replay or cannot save it in time, then just wait until your next shot loads. Click on MENU and choose REPLAY LAST SHOT. From there you'll be able to save it and even share it with friends and such.
Oh and you can save up to 10 replays at any time.
Good Luck =)