What is a good way to catch the Mysterious Stranger?

He shows up in random places, so there is no place he usually shows up.

One hint is the volume of the music. If the music is louder, he's closer to where you're currently looking. Wearing headphones helps to determine how loud it is.

Have Less Rooms

As the Mysterious Stranger can appear in any room, you'll increase your chances of spotting him if you decrease the number of places he can spawn. Get rid of excess items and decrease your storage rooms (because you don't really need 100 things of Yarn or those 18 .32's), upgrade your rooms to max so you can run less off them, make your Vault as compact as possible to also get rid of elevators.

Use Color-Clashing Rooms

With the new feature of theme rooms, choose themes for your rooms that are not brownish, so that when he does show up, he will be easier to spot.

Give All Your Dwellers Highly Visible Outfits

Just how it's harder to spot him in a brown room, it will be harder to spot him when all your dwellers are wearing trenchcoats.

Get A Pet

A specific type of tiger-striped cat will increase the chance of him spawning. I have a cat named Kato that gives a 7.5x bonus to spawns!

Take Screenshots

As @psycho brm has stated, take a screenshot once you know he's there, and play a little game of Where's the Mysterious Stranger before opening up the app again and spotting him.

Other "Strageties"

I'm of the mindset that he usually spawns just outside of the visible space you are currently looking at in-game, but for every time I think I can confirm this hypothesis, there are 2 or more times when this doesn't work.

Make a screenshot (with HD zoom out disabled), switch to photos application to find him, then come back to get him.

Turn on vibration in settings and you can do this everytime the phone vibrates, while watching movie or something...