If you have count data with small numbers and a bunch of 0's what would be the best distribution to model this? Is Poisson good at handling data with a bunch of 0's and small number between 1 and 4?
It might be Poisson, or, if there are really a LOT of zeroes, possibly zero inflated Poisson. More general options are negative binomial and zero inflated negative binomial. It's hard to tell just from what you've said, but I am guessing the NB isn't needed.
A two-part regression may apply here as well - a logistic regression that predicts whether there are zero or 1+ counts, and a second Poisson or NB regression predicting the number of non-zero counts.