Yes, from time to time, folks will ask to see your log. It doesn't happen often but when it does I'm usually impressed - it means that they're willing to lose the business rather than take an underqualified diver along and that's a good thing.
I've needed my log for the Bonne Terre folks plus operators in the Keys, Hawaii, Bonaire, Tobermorey and several training agencies. Nobody has ever beefed about occasionally brief entries (rarely signed by anyone, for instance) or water-stained pages. They're looking for different things - how many dives recently, how many dives in similar conditions, how many dives, what cool places have I been, etc. So long as you don't take it personally, it's a nice way to start a conversation and put everyone at ease.
I always have my log with me when I'm diving. In addition to my log entries, it contains emergency information plus copies of certification cards and travel documents in case the originals go astray.