A Model 37 was my first gun and I still have mine, a .410 with a 28" full choke barrel. They DO hold up well; I got mine in 1951 and it's been shot and hunted with almost every small game season since then, mostly with the AWESOME 3" magnum shells. Great bunny buster and good for "things that go bump in the night" like coons and possums.
The stock fit and barrel patterning is as good as any field grade Winchester. The finish, especially inside, isn't.
I disabled the ejector on my .410 early on, since it would loft a pricey loaded .410 shell into the mud every time. Or into a cow pie.
I used a 16 guage 37 with a 28" barrel for several bird seasons for lack of anything better and it killed flying birds quite well, (I shot a lot of spreaders!).
Winchester made more 37s over the years than any other shotgun model. Most weren't .410s--small bore ones are harder to find than 12s and 16s, with the latter most common in Latin America where some have 36" barrels.
I've never seen one with a serial number, although the post-1968 ones are supposed to have them.