BRST operators for two-dimensional theories with spin-2 and spin-s currents, generalising the W3 BRST operator of Thierry-Mieg, have previously been obtained. The construction was based on demanding nilpotence of the BRST operators, making no reference to whether or not an underlying W algebra exists. In this paper, we analyse the known cases (s=3, 4, 5 and 6), showing that the two s=4 BRST operators are associated with the WB2 algebra, and that two of the four s=6 BRST operators are associated with the WG2 algebra. We discuss the cohomology of all the know higher-spin BRST operators, the Weyl symmetry of their physical states, and their relation with certain minimal models. We also obtain the BRST operator for the case s=7. 1994.
