  In this paper, we consider the following question: how many degree $d$ curves
are there in $\mathbb{P}^3$ (passing through the right number of generic lines
and points), whose image lies inside a $\mathbb{P}^2$, having $\delta$ nodes
and one singularity of codimension $k$. We obtain an explicit formula for this
number when $\delta+k \leq 4$ (i.e. the total codimension of the singularities
is not more than four). We use a topological method to compute the degenerate
contribution to the Euler class; it is an extension of the method that
originates in a paper by A. Zinger and which is further pursued by S. Basu and
the second author. Using this method, we have obtained formulas when the
singularities present are more degenerate than nodes (such as cusps, tacnodes
and triple points). When the singularities are only nodes, we have verified
that our answers are consistent with those obtained by by S. Kleiman and R.
Piene and by T. Laarakker. We also verify that our answer for the
characteristic number of planar cubics with a cusp and the number of planar
quartics with two nodes and one cusp is consistent with the answer obtained by
R. Singh and the second author, where they compute the characteristic number of
rational planar curves in $\mathbb{P}^3$ with a cusp. We also verify some of
the numbers predicted by the conjecture made by Pandharipande, regarding the
enumerativity of BPS numbers for $\mathbb{P}^3$.
