As said in Alex's answer, smoothness is too coarse (=automatic) for classification of finite objects. It concerns equivalence relations on uncountable sets (borel spaces).
On the other hand, there has been great progress on the classification of $p$-groups lately. You can try this 2008 presentation by Bettina Eick (she also has a paper in a 2008 LMS Bulletin whith Leedham-Green, by I dont't have a link).