If you want to paste the contents of a named register without exiting Insert mode, hit <C-r>, and then the register name.

For example, <C-r>1 will paste the text that was most recently deleted, <C-r>0 will paste the text that was most recently yanked, <C-r>q will paste the contents of the q macro, etc.

This also works while in Comand-line mode, too:

:%s/(... hit <C-r>0 to paste in yanked text to replace ...)/<replacement text>/g

Alternately, you can just do <C-o>p to paste from within Insert mode. (Hitting <C-o> will temporarily bring you to Normal mode for one command and dropping you back into Insert mode right after.)

See :help registers for more information on what the various registers contain.