  In this work, we introduce the notion of local and $2$-local
$\delta$-derivations and describe local and $2$-local $\frac{1}{2}$-derivation
of finite-dimensional solvable Lie algebras with filiform, Heisenberg, and
abelian nilradicals. Moreover, we describe the local $\frac{1}{2}$-derivation
of oscillator Lie algebras, Schr{\"o}dinger algebras, and Lie algebra with a
three-dimensional simple part, whose radical is an irreducible module. We prove
that an algebra with only trivial $\frac{1}{2}$-derivation does not admit local
and $2$-local $\frac{1}{2}$-derivation, which is not $\frac{1}{2}$-derivation.
