Are There Many Artists Who Write & Sing Their Own Songs Like The Beatles Did?

There are plenty of artists that write and well as perform their own songs...unfortunately most of the artists on the radio do have any part in their own music. If you are looking for some original stuff try looking for indie music. Itunes usually has good artists and you can usually find some decent stuff on myspace as well. There are plenty of great musicians out there...you just need to search for them.

Yes there is for example prince

Yes there are quite a few artists who do that but they are obviously less than the other artists who only sing songs written by some other writers. Guns and Roses, U2, Roxen are some examples.