Were dinosaurs with feathers common?
Keep in mind that although I am a paleontologist, I am not a vertebrate paleontologist so I might not be aware of every single find ever made of feathered dinosaur.  

As far as I can tell, all dinosaurs fossils exhibiting feathers belong to the Theropoda. Here is a (probably dated) phylogeny of dinosaurs from Sereno 1999:


Credit: Paul C. Sereno

Theropoda is the group from Eoraptor to Euornithes. 

While Eoraptor  is Triassic, the group diversifies mostly during the Jurassic (still Sereno 1999) and it is also during that time that we start having "common" feathered dinosaur fossil evidence (e.g. Zhang et al. 2008, Sullivan et al. 2014).
Birds (Aves) seem to have appeared in the late Jurassic / early Cretaceous as well, but modern birds (Neornithes), i. e. birds with living representative, appeared only in the late Cretaceous.  

I don't know of any proper (sample-standardized, with coherent bias detection and removal) paleodiversity study for the theropods, although there is one for Aves (Brocklehurst et al. 2012), so it's hard to discuss about it at a species level.

In theory, feathers are more likely to fossilize than other soft tissue (Davis &amp; Briggs 1995) but in practice they are mostly found in finely grained sediments such as limestones, thus limiting the amount of outcrops in which they can be found.

Sources:
Brocklehurst, N., Upchurch, P., Mannion, P. D., O'Connor, J., 2012. The Completeness of the Fossil Record of Mesozoic Birds: Implications for Early Avian Evolution. PLoS ONE 7(6): e39056.
Davis, P. G., Briggs, D. E. G., 1995. Fossilization of feathers. Geology, 23(9): 783-786.
Sereno, P., 1999. The evolution of Dinosaurs. Science, 284: 2137-2147.
Sullivan, C. et al. 2014. The vertebrates of the Jurassic Daohugou Biota of northeastern China. Journal of Vertebrate Paleontology, 34: 243-280.
Zhang, F. et al.  2008. A bizarre Jurassic maniraptoran from China with elongate ribbon-like feathers. Nature, 455: 1105-1108.