The Nash problem on arcs for normal surface singularities states that there are as many arc families on a germ (S,O) of a singular surface as there are essential divisors over (S,O). It is known that this problem can be reduced to the study of quasi-rational singularities. In this paper we give a positive answer to the Nash problem for a family of non-rational quasi-rational hypersurfaces. The same method is applied to answer positively to this problem in the case of E_6 and E_7 type singularities, and to provide new proof in the case of D_n, n> =4, type singularities.