Due to a messed up update (misconfiguration of Gnome), I was forced to reinstall my Linux. I took this opportunity to upgrade to LMDE 201303 (Update pack 6). When starting up the new system everything worked fine, but as I went along to configure it as I wanted it to be, Firefox and Thunderbird stopped working. Firefox did not start anymore at all, without any error message. Thunderbird tried to send a crash report, but failed also without any error message.
It took me some iterations of reinstalling the system to solve the mystery. So in case you stumble into the same problem, here is my cause/solution:
I had my system set up with different partitions, so they had to be mounted. I usually use ( /, /usr, /opt and /home). As it turned out everything fell apart as soon as I mounted my old /opt partition. The reason for this is that both Firefox and Thunderbird are installed there (on Linux Mint Debian Edition at least). So this of course meant that The configuration and the application did not match.