I have a Samsung Galaxy S2. I uninstalled all apps I had recently installed and then uninstalled and reinstalled the Skype app, but this has not solved the problem. Just for the heck of it, I just tried changing my Skype password to see whether this makes any difference. If it magically helps, I'll post back in a day, otherwise assume it didn't. :-/
As expected, the password change did not have an effect.
However, I was on my service provider's data network (Virgin mobile in Canada) for half of yesterday and Skype did not sign out in that time. When I returned home in the evening and switched to wifi, Skype signed out within an hour. Thinking back, this has so far only happened while I've been on wireless networks (it also happened at work, so it is not just an artifact of my home network). When have the others had trouble?
BTW, I have Android 2.3.3.
FYI, there is another thread where many have recently complained of the same problem, no fix there yet:
Skype ver. 126.96.36.1990
Android ver. 2.3.5
Kernel ver. 188.8.131.52
Samsung Galaxy Y (GT-S5360L) on WiFi only.