In mysql-server-5.1.71-1.el6.x86_64 there is a check in init script:

[ -S /var/lib/mysql/mysql.sock ]

When the socket exist (unclean shutdown for example), the mysqld did not start because this check by printing a message that this socket is in use, which is wrong. There should be check by lsof instead:

# lsof /var/lib/mysql/mysql.sock 
COMMAND   PID  USER   FD   TYPE             DEVICE SIZE/OFF   NODE NAME
mysqld  14235 mysql   12u  unix 0xffff880214ec0980      0t0 169827 /var/lib/mysql/mysql.sock

Only lsof could tell if the socket is really in use or not. If yes, the output above tells. If the socket is not in use, the output above is empty.

The workaround is to delete the socket manually before trying to start MySQL daemon.