Well you certainly asked the right question. If the Sylow subgroup is cyclic then there is a nice way to write down all the (finitely many) indecomposable modules, see for example the last chapter of Alperin's "Local Representation Theory" or 6.5 in Benson's "Representations and Cohomology." If $p$ is odd and one is not in the cyclic defect case then the representation type is wild, and classifying the indecomposable modules is hopeless in some precise sense that it is at least as difficult as for a polynomial algebra on two noncommuting variables.

In between these two is something called "tame" representation type, which happens for symmetric groups only for $p=2$ and $n=4, 5$, where there is a dihedral Sylow. (Or also for larger $n$ but but blocks of weight $2$). However by work of Jost it is known that all these blocks are Morita equivalent to either the principal block of $S_4$ or $S_5$. The main reference for this situation is Erdmann's LNM 1428 although there is a lot of work since. See also for example the paper "Representation type of Hecke algebras of type A" by Erdmann-Nakano that gives the quiver and relations for the principal blocks of $S_4$ and $S_5$ in characteristic two.

As an aside, even in the wild type case I find it an interesting question to ask if one can classify indecomposable, self-dual modules that have Specht filtrations.