There is a lot of material available about the representation theory of the symmetric group over $\mathbb{C}$ and fields of characteristic $0$. In particular, there is the decomposition of the group algebra $$ \mathbb{C}S_n = \bigoplus_{\lambda \vdash n} S^{\lambda} $$ where the $S^{\lambda}$ are the Specht modules, which are irreducible and pairwise disjoint over $\mathbb{C}$.
All the resources I could find ignored the characteristic $p > 0$ case, or said that it was hard in general when $p\vert n!$, but what of the (simple?) case when $p = 2$?
What is known about the decomposition of $\mathbb{F}_2S_n$ or $\overline{\mathbb{F}}_2S_n$ into irreducible modules, particularly the number of irreducible components?
The textbook by James–Kerber (1981) has lots of the (then) recent developments and is a standard reference. Kleshchev (2005) has more recent developments.
Here are the composition lengths of the Specht modules for the first 8 symmetric groups: $$\scriptsize\begin{array}{r|rr} n& \\\hline 1 & 1 \\ 2 & 1 & 1 \\ 3 & 1 & 1 & 1\\ 4 & 1 & 2 & 1 & 2 & 1\\ 5 & 1 & 1 & 2 & 3 & 2 & 1 & 1\\ 6 & 1 & 2 & 3 & 2 & 4 & 1 & 2 & 4 & 3 & 2 & 1\\ 7 & 1 & 1 & 1 & 2 & 2 & 3 & 2 & 2 & 3 & 3 & 2 & 2 & 1 & 1 & 1\\ 8 & 1 & 2 & 2 & 3 & 2 & 3 & 1 & 6 & 4 & 3 & 5 & 8 & 4 & 6 & 2 & 5 & 1 & 3 & 3 & 2 & 2 & 1\\ \end{array} $$
Here are the max composition lengths (the trivial rep always has CS length 1)
$$\scriptsize \begin{array}{rrrrrrrrrrrrrrrrr} n&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19\\\hline MaxCL&1&1&1&2&3&4&3&8&8&12&12&33&33&59&47&128&131&216&217\\ \end{array} $$
The composition lengths still have a wide range, though not as bad as previously indicated.
You can ask GAP for any particular symmetric group up to $n \leq 19$ using
List(DecompositionMatrix(CharacterTable("Symmetric",n) mod 2),Sum)The partitions corresponding to each entry are given in
List(CharacterParameters(CharacterTable("Symmetric",n)),x->x[2])Old wrong answer:
The following are the lists of composition lengths of the projective covers of the irreducible $\mathbb{F}_2[S_n]$ modules.
$$\scriptsize\begin{array}{r|rr} n & \\ \hline 1 & 1 \\ 2 & 2 \\ 3 & 2 & 1\\ 4 & 4 & 3\\ 5 & 6 & 3 & 2\\ 6 & 12 & 6 & 6 & 1\\ 7 & 8 & 3 & 6 & 6 & 4\\ 8 & 20 & 9 & 18 & 14 & 6 & 2\\ 9 & 32 & 8 & 9 & 9 & 18 & 6 & 8 & 4\\ 10 & 56 & 24 & 22 & 44 & 18 & 3 & 6 & 12 & 12 & 1\\ 11 & 60 & 32 & 17 & 24 & 26 & 12 & 18 & 24 & 22 & 12 & 5 & 8\\ 12 & 156 & 108 & 47 & 60 & 84 & 18 & 48 & 8 & 30 & 36 & 26 & 10 & 6 & 4 & 2\\ \end{array}$$ $$ %\tiny %\begin{array}{r|rr} %13 & 264 & 84 & 49 & 92 & 35 & 24 & 76 & 32 & 42 & 70 & 24 & 12 & 16 & 30 & 24 & 12 & 10 & 10\\ %14 & 480 & 248 & 240 & 130 & 80 & 204 & 20 & 28 & 142 & 48 & 40 & 80 & 80 & 24 & 36 & 24 & 3 & 14 & 24 & 5 & 20 & 6\\ %15 & 376 & 240 & 112 & 96 & 96 & 134 & 96 & 80 & 48 & 78 & 96 & 48 & 56 & 20 & 20 & 248 & 124 & 108 & 65 & 32 & 18 & 16 & 24 & 30 & 15 & 30 & 1\\ %16 & 916 & 876 & 672 & 207 & 432 & 72 & 268 & 384 & 120 & 374 & 168 & 136 & 108 & 268 & 28 & 144 & 14 & 63 & 40 & 54 & 84 & 30 & 84 & 28 & 8 & 60 & 54 & 30 & 4 & 6 & 10 & 6\\ %17 & 1560 & 320 & 780 & 261 & 240 & 304 & 432 & 269 & 372 & 208 & 384 & 134 & 120 & 324 & 80 & 112 & 164 & 60 & 132 & 190 & 48 & 116 & 66 & 40 & 108 & 96 & 96 & 56 & 30 & 51 & 48 & 20 & 30 & 60 & 20 & 42 & 15 & 2\\ %18 & 2696 & 884 & 2120 & 640 & 224 & 1256 & 416 & 736 & 124 & 392 & 1100 & 160 & 376 & 28 & 326 & 108 & 340 & 68 & 80 & 618 & 236 & 316 & 184 & 24 & 168 & 71 & 280 & 30 & 60 & 196 & 174 & 30 & 100 & 20 & 192 & 84 & 14 & 68 & 12 & 34 & 40 & 15 & 68 & 15 & 5 & 20\\ %19 & 3976 & 1340 & 932 & 1168 & 908 & 720 & 960 & 800 & 536 & 264 & 680 & 368 & 504 & 140 & 590 & 460 & 100 & 288 & 132 & 354 & 562 & 444 & 432 & 120 & 80 & 547 & 252 & 154 & 128 & 108 & 136 & 160 & 128 & 360 & 228 & 100 & 252 & 60 & 70 & 134 & 6 & 80 & 81 & 148 & 66 & 64 & 52 & 30 & 40 & 84 & 3 & 25 & 40 & 15\\ %\end{array} $$
There seems to be a pretty wide range.