Let
π
:
Y
→
X
\pi \colon Y\to X
denote the canonical resolution of the two dimensional Kleinian singularity
X
X
of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of
ω
\omega
-semistable properly supported sheaves on
Y
Y
with fixed slope
μ
\mu
and
ζ
\zeta
-semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on
ζ
\zeta
depending on the polarization
ω
\omega
and
μ
\mu
. These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide a finer description of the cohomological, K-theoretical and categorified Hall algebra of
ω
\omega
-semistable properly supported sheaves on
Y
Y
with fixed slope
μ
\mu
: for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams.