  We present a convenient method of algebraic classification of 2+1 spacetimes
into the types I, II, D, III, N and O, without using any field equations. It is
based on the 2+1 analogue of the Newman-Penrose curvature scalars $\Psi_A$ of
distinct boost weights, which are specific projections of the Cotton tensor
onto a suitable null triad. The algebraic types are then simply determined by
the gradual vanishing of such Cotton scalars, starting with those of the
highest boost weight. This classification is directly related to the specific
multiplicity of the Cotton-aligned null directions (CANDs) and to the
corresponding Bel-Debever criteria. Using a bivector (that is 2-form)
decomposition, we demonstrate that our method is fully equivalent to the usual
Petrov-type classification of 2+1 spacetimes based on the eigenvalue problem
and determining the respective canonical Jordan form of the Cotton-York tensor.
We also derive a simple synoptic algorithm of algebraic classification based on
the key polynomial curvature invariants. To show the practical usefulness of
our approach, we perform the classification of several explicit examples,
namely the general class of Robinson-Trautman spacetimes with an aligned
electromagnetic field and a cosmological constant, and other metrics of various
algebraic types.
