  Work of the last few years has shown that the key algebraic features of
Logarithmic Conformal Field Theories (LCFTs) are already present in some finite
lattice systems (such as the XXZ spin-1/2 chain) before the continuum limit is
taken. This has provided a very convenient way to analyze the structure of
indecomposable Virasoro modules and to obtain fusion rules for a variety of
models such as (boundary) percolation etc. LCFTs allow for additional quantum
numbers describing the fine structure of the indecomposable modules, and
generalizing the `b-number' introduced initially by Gurarie for the c=0 case.
The determination of these indecomposability parameters has given rise to a lot
of algebraic work, but their physical meaning has remained somewhat elusive. In
a recent paper, a way to measure b for boundary percolation and polymers was
proposed. We generalize this work here by devising a general strategy to
compute matrix elements of Virasoro generators from the numerical analysis of
lattice models and their continuum limit. The method is applied to XXZ spin-1/2
and spin-1 chains with open (free) boundary conditions. They are related to
gl(n+m|m) and osp(n+2m|2m)-invariant superspin chains and to nonlinear sigma
models with supercoset target spaces. These models can also be formulated in
terms of dense and dilute loop gas. We check the method in many cases where the
results were already known analytically. Furthermore, we also confront our
findings with a construction generalizing Gurarie's, where logarithms emerge
naturally in operator product expansions to compensate for apparently divergent
terms. This argument actually allows us to compute indecomposability parameters
in any logarithmic theory. A central result of our study is the construction of
a Kac table for the indecomposability parameters of the logarithmic minimal
models LM(1,p) and LM(p,p+1).
