How can tell that my last shutdown was properly done in Ubuntu 11.10 or if the machine was powered off before shutting down the OS?
If the machine was shutdown properly then there must be a shutdown log logged in kern.log file in /var/log directory. After a shutdown whenever a normal boot occurs the OS writes the log for the same in kern.log. Hence every boot log must be preceded by a shutdown log if the booting and shutdown process was normal.
Whenever a normal shutdown occurs "Kernel logging (proc) stopped." is written in kern.log. Similarly whenever a boot occurs "imklog 5.8.1, log source = /proc/kmsg started." is written in kern.log
These two messages should be in the order if the shutdown was normal rather than abrupt power off. There should be no "imklog 5.8.1, log source = /proc/kmsg started." message without the "Kernel logging (proc) stopped." message if the previous shutdown was normal. The both message should always occur in pair in log.
Just type in terminal :- gedit /var/log/kern.log
and check for the shutdown and boot log pairs. If they are found not in pairs anywhere that shutdown must have been abrupt.