Just tried installing Designer on my Ubuntu 10.04, 64 bit desktop. The desktop shortcut did no work, and so tried running from the command line. Got this error:
mv: cannot stat `/home/david/.config/Trolltech.conf': No such file or directory
Loading libion: /home/david/.local/share/data/Sencha, Inc./Sencha Designer 1.2/libion-1.2.0.so
Grrr right now we don't support 10.04 64-bit. We do support 11.04 which may or may not help you! I wish we had a build engineer so we could get support for all distros. At some point we'll need one. You of course can also run a virtual machine running say 11.04 or 11.10 of Ubuntu.
I had the same issue running the .bin file on Ubuntu 11.10 and the segfault on the other file (though that was not identical, probably because I already had some of the QT libraries installed). For me the fix was simple:
sudo apt-get install libqtwebkit4
Again, I'm running 11.10, so your mileage may vary.