I'll talk about the ones I've handled or shot or owned.

Chapuis: As most people know I've a little 9.3 that I've owened, hunted with and shot for over 5 or so years. I've also handled a fair ammount of them and visited their factory back in 2001. I also have friends in Europe who use them every weekend for boar hunting

Pro: nice looking rifles (except that blasted beavertail on the big bores...), nice handling, and a good price (especialy if you buy in Europe). Normally they are well regulated. The double underlug design is also supposed to be very strong and durable, but I am not an expert so I can not say for sure. Also they are almost fully customizable when you order a new one: stock measurements, stock styles, barrel length, sights, engraving, etc. Also you can make requests for the factory load you want the rifle regulated for.

Con: Customer service could really be improved on. Also like many of the "factory" doubles (like Aya, etc), they can have quality control issues especially on finishing details. My top lever catch was not finished correctly on my new rifle. But hey, when I bought mine they were only selling for about $3,000 brand new so what can you expect...

Merkel: Handled many of them, and shot a 9.3 and 500NE. Also I have several friends who use them every weekend for boar hunting.
Pro: good solid rifle for the money. The ones I've shot were all well regulated (a 9.3 and a 500NE). They look to be solidly built, and the ones I've seen are very tight when new which seems to be a good sign of being solid (at least to my limited eyes).

Con: Their importer here in the US, GSI sucks and rips people off on the prices they charge. Like all Merkels they have a certain "Teutonic style" that some of the Anglo-philes don't like. You can not customized much with a factory order (at least that's my understanding). Also though I've never personally seen a Merkel double rifle with the quality control issues that I've mentioned about Chapuis, I have seen some Merkel shotguns with quality control issues: bad wood to metal finish, creepy triggers, etc. Not a big deal, but I think standard with the "factory produced" doubles whether shotguns or rifles. Also I've heard many people complain about the factory sights and the buttplate, but thats just what I've heard second hand.

Heym: (only handled several of them)
Pro: They appear to be a higher quality rifle than either Chapuis or Merkel. I've also heard a few gunsmith friends of mine say that they are much better regulated than them either, but I've no proof of that.

Con: Style wise there is something even more "Teutonic" about them than Merkel. Many people say they are ugly, but I don't quite agree with that. Also they are significantly more expensive than the others (I've seen them for twice the price).

I like both Merkel and Chapuis and once they've been looked over by a good smith I think they make great utility double rifles.