  We study the irreducible components of the moduli space of instanton sheaves
on $\mathbb{P}^3$, that is rank 2 torsion free sheaves $E$ with
$c_1(E)=c_3(E)=0$ satisfying $h^1(E(-2))=h^2(E(-2))=0$. In particular, we
classify all instanton sheaves with $c_2(E)\le4$, describing all the
irreducible components of their moduli space. A key ingredient for our argument
is the study of the moduli space ${\mathcal T}(d)$ of stable sheaves on
$\mathbb{P}^3$ with Hilbert polynomial $P(t)=d\cdot t$, which contains, as an
open subset, the moduli space of rank 0 instanton sheaves of multiplicity $d$;
we describe all the irreducible components of ${\mathcal T}(d)$ for $d\le4$.
