  In three dimensions, it is known that field theories possessing extended
$(p,q)$ anti-de Sitter (AdS) supersymmetry with ${\cal N}=p+q \geq 3$ can be
realised in (2,0) AdS superspace. Here we present a formalism to reduce every
field theory with (2,0) AdS supersymmetry to ${\cal N}=1$ AdS superspace. As
nontrivial examples, we consider supersymmetric nonlinear sigma models
formulated in terms of ${\cal N}=2$ chiral and linear supermultiplets. The
$(2,0) \to (1,0)$ AdS reduction technique is then applied to the off-shell
massless higher-spin supermultiplets in (2,0) AdS superspace constructed in
[1]. As a result, for each superspin value $\hat s$, integer $(\hat s= s)$ or
half-integer $(\hat s= s+1/2)$, with $s=1,2,\dots $, we obtain two off-shell
formulations for a massless ${\cal N}=1$ superspin-$\hat s$ multiplet in
AdS${}_3$. These models prove to be related to each other by a superfield
Legendre transformation in the flat superspace limit, but the duality is not
lifted to the AdS case. Two out of the four series of ${\cal N}=1$
supersymmetric higher-spin models thus derived are new. The constructed
massless ${\cal N}=1$ supersymmetric higher-spin actions in AdS${}_3$ are used
to formulate (i) higher-spin supercurrent multiplets in ${\cal N}=1$ AdS
superspace; and (ii) new topologically massive higher-spin off-shell
supermultiplets. Examples of ${\cal N}=1$ higher-spin supercurrents are given
for models of a complex scalar supermultiplet. We also present two new
off-shell formulations for a massive ${\cal N}=1$ gravitino supermultiplet in
AdS${}_3$.
