  This paper is mainly concerned with the construction of new off-shell higher
spin N=1 supermultiplets in three spacetime dimensions. We elaborate on the
gauge prepotentials and linearised super-Cotton tensors for higher spin N=1
superconformal geometry and propose compensating superfields required to
formulate off-shell massless higher spin supermultiplets. The corresponding
gauge-invariant actions are worked out explicitly using an auxiliary oscillator
realisation. We construct, for the first time, off-shell massive higher spin
supermultiplets. The gauge-invariant actions for these supermultiplets are
obtained by adding Chern-Simons like mass terms (that is, higher spin
extensions of the linearised action for N=1 conformal supergravity) to the
actions for the massless supermultiplets. For each of the massive gravitino and
supergravity multiplets, we propose two dually equivalent formulations.
