  We consider a nonlinear Dirichlet problem driven by a nonhomogeneous
differential operator with a growth of order $(p-1)$ near $+\infty$ and with a
reaction which has the competing effects of a parametric singular term and a
$(p-1)$-superlinear perturbation which does not satisfy the usual
Ambrosetti-Rabinowitz condition. Using variational tools, together with
suitable truncation and strong comparison techniques, we prove a
"bifurcation-type" theorem that describes the set of positive solutions as the
parameter $\lambda$ moves on the positive semiaxis. We also show that for every
$\lambda>0$, the problem has a smallest positive solution $u^*_\lambda$ and we
demonstrate the monotonicity and continuity properties of the map
$\lambda\mapsto u^*_\lambda$.
