The Maritime Museum in Aberdeen is brilliant. Not nearly on the same scale as some of the entries in this thread, but it's awesome.

It's modern, it's clean, and it's free. It's quite clearly funded by the oil and gas industry (and it doesn't hide from it where exhibits have been financed) but it's an absolutely fantastic way to spend an afternoon - from the introduction of maritime and naval operations in Scotland to present day works in the North Sea - and a guest hall where seasonal exhibits are hosted.

Of particular note is the recreation of the Piper Alpha cabins and the recovered life vests from that fateful night - genuinely interesting and a museum that punches well above it's weight.