In this paper, we establish general stratawise higher jet evaluation transversality of J-holomorphic curves for a generic choice of almost complex structures J (tame to a given symplectic manifold (M, $omega$)). Using this transversality result, we prove that there exists a subset $cal{J}^{ram}_{omega};{subset};cal{J}_{omega}$ of second category such that for every $J;{in};cal{J}^{ram}_{omega}$, the dimension of the moduli space of (somewhere injective) J-holomorphic curves with a given ramication prole goes down by 2n or 2(n - 1) depending on whether the ramication degree goes up by one or a new ramication point is created. We also derive that for each $J;{in};cal{J}^{ram}_{omega}$ there are only a finite number of ramication profiles of J-holomorphic curves in a given homology class $eta;{in};H_2$(M; $mathbb{Z}$) and provide an explicit upper bound on the number of ramication proles in terms of $c_1(eta)$ and the genus g of the domain surface.