Our hotel, Albergo Cesàri, was quite near the Trevi Fountain and thus we passed by it quite a few times. It always had large crowds - day or night. It was made famous in the movie - Three Coins in the Fountain. I read in Wikipedia that the song with the same name was composed in an hour where the authors hadn't seen the movie or the script. The song got the Best Song Academy Award in 1954.
There were lots of coins in the fountain. We saw several instances of people with some form of fishing device retrieving coins one a time. I didn't see anyone jump in.
Click here for some pictures - the first few are the Trevi Fountain: