Don't do ANYTHING right now. Bide your time and think this through.
Let him think you are totally on his side and indignant about Gossip Bloke.
I'd possibly leave it a full week or so, just make supportive noises - 'by the way he hasn't contacted me after all, he'd be stupid to with that ridiculous story but I'll let you know if he does darling!'
Give him time to relax about it. Right now, if you snoop he will be on high alert. He has probably cooled things right now, if it is an affair, just in case. It's the worst time to go digging.
What is the situation - how far is the work/MIL location from you? How old are your kids and how free are you - could you get over there and snoop, for example? How is he with his phone (however a second phone is possible anyway). How often do you speak to him when he's away - do you have fixed times, or might he expect you to call randomly? Does he not answer sometimes when you call?
You can check out the MIL story fairly easily. You'll have to lie low until he relaxes though. Pick a night you know he's supposed to be there, just through casual conversation. Then wait until late evening and call the land line from your land line or call box - say you've lost your phone and had MIL's number on a bit of paper but not his. Invent something you need to contact him for - leaking pipe or electrics gone off or fence blown down, does he have the number of hte usual plumber or whatever (you've lost your phone, remember?) Or even a call to let him know you've lost your phone so don't panic if it's not answered. That's all quite plausible. And if MIL answers and he's not there - 'Oh, do you know when you expect him? Should he be with you tonight - I'm not really sure what he's doing - will he be with you tomorrow if not tonight? As long as I can get hold of him by tomorrow... If the response is bafflement, no darling I'm not really expecting him to stay this week, did he say he was? - then you have him.
But don't do anything for a good week or so.