Do singularities of plane curves deform independently? If $C\subset\mathbb{P}^2$ is an integral curve of degree $d$, do its singularities deform independently as we vary $C$ over degree $d$ curves? If not, what about in the case $C$ is a nodal curve or $C$ is a general nodal curve?
Edit: In other words, is the composition $H^0(C,N_{C/\mathbb{P}^2})={\rm Hom}(N_{C/\mathbb{P}^2}^\vee,\mathscr{O}_C)\rightarrow {\rm Ext}^1(\Omega_C,\mathscr{O}_C)\rightarrow Ext^1(\Omega_C,\mathscr{O}_C)$ surjective? 
(If I did the reduction correctly, which is not a given, I think this is equivalent to the vector space $V=k[X,Y,Z]_d$ of degree $d$ homogenous polynomials surjecting onto the sheaf associated to the graded module $k[X,Y,Z]/(F,\partial_XF,\partial_YF,\partial_ZF)$, where $C=V(F)$.)
 A: A good survey on this problem is the paper by Greuel, Lossen and Shustin
Equisingular families of projective curves, see http://arxiv.org/pdf/math/0612310.pdf.
In particular, at page 5 one can find the reference [Sev68] to the work of Severi showing that singular points of a nodal curve,
irreducible or not, can be smoothed, or preserved, independently. This answers the last part of your question.  
Regarding the possibility of independently smoothing singularities worse than nodes, as explained in J. Starr's comment the answer is in general no, even if this can be hard to check in explicit examples.
In fact, at p. 30 of the aforementioned paper the authors state the following more general question for cuspidal curves:

Are there $k$ and $d$ such that a curve of degree $d$ with $k$ cusps does exist, but with $k′ < k$ cusps does not ?

A candidate are a series of irreducible cuspidal curves constructed by Hirano (reference [Hir92]). They have degree $d= 2 \cdot 3^t$, with $t \geq 1$, and contain precisely $k=9(9t −1)/8$ cusps.  
