Yesterday it was working fine when I turned it off, but when I booted up
the PC this morning my LCD monitor (a 15" Dell E151Fpp) was completely
dead - even the little light indicating 'power' was off.

How old is it? Where did you purchase it from?

Note that 1 year guarantees are in addition to your statutory rights. Your
statutory rights are usually considerably more extensive that the
manufacturer's guarantee. However, claims are against the retailer, not the
manufacturer. If the retailer is bust/lost/dodgy market trader etc., then
this won't be helpful and you can't pursue the manfacturer instead.

Christian.