It's not surprising to see the backup files messages.1 etc as these are created by the logrotate daemon, which doesn't know anything about syslog.
It sounds like you have a configuration problem in syslog itself; take a look at your /etc/syslog.conf file and check that it has an entry for /var/log/messages. This will vary between distributions but it should look something like this:
(line taken from a random FC5 machine)
If the line starts with a # sign then it's been commented out (in which case, remove the # sign, save the file and restart syslogd).
If you want to know more about configuring syslog then type
Feel free to post the file if you don't know what's wrong.
Hope that helps,
—Robert J Lee