- check /var/log/syslog for errors
- look especially for filesystem problems
- check dmesg output
- check uptime output (long time means stable (at least at kernel level), short time does not mean much because it could be a brand new system.
- check top output, see if it's swapping a lot, check if it's going to run out of virtual memory soon.
stress test the system (NB don't do this in production, only in pre-production environment).
- look at application logs to see if it gets major problems that can be caused by underlying OS
- etc ...