Here are some recent examples from my work.
Apache wouldn't start properly
A 'ps -def | grep http' would show only the main httpd process, with no worker processes. The Apache error log showed these lines:
Digest: generating secret for digest authentication
A google search for this line revealed this article:
It turns out the randomness/entropy on that box had been exhausted. I grabbed the rng-tools tar.gz from sourceforge, compiled and installed it, then ran
rngd -r /dev/urandom
...and apache started its worker processes instantly.
Cannot create InnoDB tables in MySQL
Here, all it took was to read the MySQL error log in /var/lib/mysql. It's very friendly indeed, and tells you exactly what to do!
InnoDB: Error: data file ./ibdata1 is of a different size
InnoDB: 2176 pages (rounded down to MB)
InnoDB: than specified in the .cnf file 128000 pages!
InnoDB: Could not open or create data files.
InnoDB: If you tried to add new data files, and it failed here,
InnoDB: you should now edit innodb_data_file_path in my.cnf back
InnoDB: to what it was, and remove the new ibdata files InnoDB created
InnoDB: in this failed attempt. InnoDB only wrote those files full of
InnoDB: zeros, but did not yet use them in any way. But be careful: do not
InnoDB: remove old data files which contain your precious data!
Windows-based Web sites are displaying errors
Many times I've seen Windows/IIS based Web sites displaying cryptical errors such as:
Server Error in '/' Application.
In conclusion -- RTFL and google it! You'll be surprised how large of a percentage of issues you can solve this way.