Who knows, does a theorem of such form exist:


Any one-parameter family of vector fields on the orientable surface may be slightly perturbed such that
1) all fields in the family except finite number are Morse-Smale;
2) at each non-Morse-Smale field a bifurcation from some list occurs (bifurcations are considered modulo topological equivalence).



Generic path in the space of vector fields on the orientable surface
The paper you are looking for is *Generic one-parameter families of vector fields on two-dimensional manifolds.* by J. Sotomayor, Inst. Hautes Études Sci. Publ. Math. No. 43 (1974), 5–46.


You cannot hope for a finite number of bifurcations. Some 1-parameter families of vector fields with infinitely many bifurcations are stable (as 1-parameter families of course). Think of infinitely many longer and longer saddle connections accumulating on a degenerate closed leave. Also you cannot hope to get rid of non-trivial recurrence by perturbation. If you can ensure absence of recurrence then the only bifurcations are what you think: parabolic singularities, weakly degenerate closed leaves and saddle connections.


As Benoît pointed out, the most relevant reference in contact topology is Giroux's paper *Structures de contact en dimension trois et bifurcations des feuilletages de surfaces*, Invent. Math. 141 (2000), no. 3, 615–689 but also the discretization lemma proved in *Sur les transformations de contact au-dessus des surfaces*, Essays on geometry and related topics, Vol. 1, 2, 329–350, Monogr. Enseign. Math., 38, Enseignement Math. (2001) which explains how to get rid of all but finitely many bifurcations through a large modification of the family. 


If you have trouble reading French, I wrote an introductory discussion of part of this in a chapter of *Contact and Symplectic Topology*, edited by Bourgeois, Colin and Stipsicz, Bolyai Society Mathematical Studies (2014). An early version can be found on arxiv. You will also find some discussion of generic singularities in the contact context in Section 2 of Thomas Vogel's *Uniqueness of the contact structure approximating a foliation*.


