Do we have any classification of all singular curves of genus $0$ and $1$ in $\mathbb{P}_2$? For example if it is of degree $2$ (a conic) then it is $L.L'$ where both $L$ and $L'$ are isomorphic to $\mathbb{P}_1$.
No. There is a lot of research focusing on singular rational plane curves and there are many open problems here. For example, it is not known how many cusps a rational irreducible plane curve can have (although it is conjectured that this number is four, see e.g. this paper by Piontkowski or Moe's thesis). I don't know what the situation is for genus 1 curves, but I would not suspect that this problem is easier. EDIT: In the above, I am assuming that you are talking about the genus of the normalization and not the arithmetic genus $p_a=(d1)(d2)/2$ (in which case the classification problem is trivial). 


Dear unknown and Ottem. What an interesting discussion. The subject of curves, and cuspidal curves in particular is by no means completely classified. The conjecture in Piontkowski's article has not been proved, but there is an article by Tono where an upper bound for the number of cusps on a cuspidal curve is given for curves of any given genus: http://onlinelibrary.wiley.com/doi/10.1002/mana.200310236/abstract 

