so at the login screen, there is the session you can choose, you have tried both kde3 and kde4 there and after logging in you get a sigsegv for both? that would surprise me a lot, since kde 3.5 should definitely still work unless something really went wrong during your other installs with adept.
which other installs? did they appear to properly configure/install?
a simple test, if you have not configured anything in your KDE4 desktop (sounds like you just installed it), then try deleting the kde4 directory. it is the directory ~/.kde4 (~ is for home directory, links to /home/username). this will not delete any files you have on the computer, just the configuration files for kde4, which are recreated to their defaults upon your next login. to be safe you could just move it, type:
control-alt-F1 then login (your at a black terminal)
type " mv ~/.kde4 ~/.kde4old "
hit control-alt-F7 to go back to the graphical UI and try logging into kde4 again. hopefully whatever cause sigsegv is gone and new defaults work.
i occasionally have had a segv when logging out, i think it's just happened when i have another session open (the control-alt-F1 terminal), so you could logout of that other session before quitting kde4 just to be sure, but i've never had any corrupted files or anything even if i have had kde4 crash once or twice.
good luck, let us know how it goes,
jet