This is a bit of a fishing expedition, because I'm not sure what I'm looking for. Very vaguely stated, here's the driving question:
What conditions on an $\omega$-stable theory make the class of countable models "well-behaved?"
Here conditions would mean model-theoretic conditions, e.g. few types, although that's a bad example since of course all $\omega$-stable theories have few types.
"Well-behaved" would mean, in terms of descriptive set theory, that the isomorphism relation for countable models of the theory is not Borel-complete. But I'd be happy even knowing when it's (say) smooth, or even when it has only countably many countable models.
Of course the Ryll-Nardzewski theorem tells us when there's exactly one countable model, but beyond that, I don't know anything in this context.
I know some results in the other direction- the conditions "eni-DOP" and "eni-deep" both imply the isomorphism relation is Borel-complete. This is a recent result of Laskowski and Shelah. But the converse doesn't seem helpful (to me); knowing (say) that the theory has eni-NDOP and is eni-shallow implies a certain technique for proving complexity won't work, but it doesn't seem to imply that models are simple, only that you'd need a new proof.
So I'm just looking for any results in the "positive" direction- some reasonable condition on the theory which implies any kind of "niceness" for the class of countable models.