  Let $(X,o)$ be a complex analytic normal surface singularity with rational
homology sphere link $M$. The `topological' lattice cohomology ${\mathbb
H}^*=\oplus_{q\geq 0} {\mathbb H}^q$ associated with $M$ and with any of its
spin$^c$ structures was introduced by the author. Each ${\mathbb H}^q$ is a
graded ${\mathbb Z}[U]$--module. Here we consider its homological version
${\mathbb H}_*=\oplus_{q\geq 0}{\mathbb H}_q$. The construction uses a
Riemann-Roch type weight function. 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 we fix the embedded topological type of a reduced curve
singularity $(C,o)$ embedded into $(X,o)$, that is, a 1-dimensional link
$L_C\subset M$. Each component of $L_C$ will also carry a non-negative integral
decoration. For any fixed $n$, the embedded link $L_C$ provides a natural
filtration of the space $S_n$, which induces 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 the decorated $(M,L_C)$. Each page provides a triple graded
${\mathbb Z}[U]$-module.
  We provide several concrete computations of these pages and structure
theorems for the corresponding multivariable Poincar\'e series associated with
the entries of the spectral sequences.
  Connections with Jacobi theta series are also discussed.
