Is there a file system for linux where it's possible (and safe) to perform a file system check while it's mounted read write or which does not need to be checked periodically ?
E.g. a file system whose check first 'grabs' the entire file system and then releases those parts which have been checked already.
(I'm looking for file systems with capabilities like ext2 or better, i.e. something I could use as a replacement for a root or /home file system on a PC)
Don't run fsck on a mounted file system. fsck on a mounted filesystem can data corruption.
btrfs:
I don't quite feel comfortable using btrfs yet. Its too new.
I've been using xfs for many years, because fsck takes seconds for any sized filesystem. The other reason is that expanding the filesystem online also takes seconds.
You could online fsck a filesystem snapshot. See my answer for a similar question.