Question

is there a relationship between commonness of a trait and whether it is dominant or recessive?

Answer 1

Yes at least in some cases

if gene gives unfavorable characteristic and its recessive then that allele will be found at very low occurrence . for example allele responsible for albunism is very rare in corn plant over norm allele

Answer 2

the simple answer is no.
brown eyes is a dominant trait that is abundant.
autosomal dominant genetic disease, even horrible like huntingtons are - thanks god very rare