  Let $(C,o)$ be a complex analytic isolated curve singularity of arbitrary
large embedded dimension. Its lattice cohomology ${\mathbb H}^*=\oplus_{q\geq
0}{\mathbb H}^q$ was introduced by \'Agoston and the author, each ${\mathbb
H}^q$ is a graded ${\mathbb Z}[U]$--module. Here we study its homological
version ${\mathbb H}_*(C,o)=\oplus_{q\geq 0}{\mathbb H}_q$. The construction
uses the multivariable Hilbert function associated with the valuations provided
by the normalization of the curve. A key intermediate product is a tower of
spaces $\{S_n\}_{n\in {\mathbb Z}}$ such that ${\mathbb H}_q=\oplus_n
H_q(S_n,{\mathbb Z})$.
  In this article for every $n$ we consider a natural filtration of the space
$S_n$, which provides a homological spectral sequence converging to the
homogeneous summand $H_q(S_n,{\mathbb Z})$ of the lattice homology. All the
entries of all the pages of the spectral sequences are new invariants of
$(C,o)$. We show how the collection of the first pages is equivalent with the
motivic Poincar\'e series of $(C,o)$.We provide several concrete computations
of the corresponding multivariable Poincar\'e series associated with the
entries of the spectral sequences.
  In the case of plane curve singularities, the first page can also be
identified with the Heegaard Floer Link homology of the link of the
singularity. In this way, the new invariants provide for an arbitrary (non
necessarily plane) singularity a homological theory which is the analogue of
the Heegaard Floer Link theory for links of plane curve singularities.
