We construct the analogue of the Heegaard Floer Link homology for an algebraic curve singularity, even if it is not planar, that is, for curve germs $(C,o)\subset ({\mathbb C}^N,o)$. In the first step we define the lattice homology of the germ. Then, via a convenient filtration we define a spectral sequence, whose $E^\infty$ page is the graded lattice homology, while the first page $E^1$ is the above mentioned invariant. If the curve singularity is in $({\mathbb C}^2,0)$ then the $E^1$ page can be identified with the Heegaard Floer Link homology of the algebraic link in $S^3$ associated with $(C,o)$; but even in this case the spectral sequence is `new' (it is not the spectral sequence from $HFL^-$ theory which converges to $HF^-(S^3)$). All the pages are well-defined invariants of the singularity (even if it is not planar).