  In three spacetime dimensions, (super)conformal geometry is controlled by the
(super) Cotton tensor. We present a new duality transformation for N-extended
supersymmetric theories formulated in terms of the linearised super-Cotton
tensor or its higher spin extensions for the cases N=2, 1, 0. In the N=2 case,
this transformation is a generalisation of the linear-chiral duality, which is
known to provide a dual description in terms of chiral superfields for general
models of self-interacting N=2 vector multiplets in three dimensions and N=1
tensor multiplets in four dimensions. For superspin-1 (gravitino multiplet),
superspin-3/2 (supergravity multiplet) and higher superspins s>3/2, the duality
transformation relates a higher-derivative theory to one containing at most two
derivatives at the component level. In the N=1 case, we introduce gauge
prepotentials for higher spin superconformal gravity and construct the
corresponding super-Cotton tensors, as well as the higher spin extensions of
the linearised N=1 conformal supergravity action. Our N=1 duality
transformation is a higher spin extension of the known superfield duality
relating the massless N=1 vector and scalar multiplets. In the
non-supersymmetric (N=0) case, the gauge prepotentials for higher spin
conformal geometry (both bosonic and fermionic) and the corresponding Cotton
tensors can be obtained from their N=1 counterparts by carrying out N=1 --> N=0
reduction. In the bosonic sector, this reduction is shown to lead to the Cotton
tensors for higher spins constructed by Pope and Townsend, and by Damour and
Deser in the spin-3 case. Our N=0 duality transformation is a higher spin
extension of the vector-scalar duality.
