  We determine restrictions on the singularity content of a Fano polygon, or
equivalently of certain orbifold del Pezzo surfaces. We establish bounds on the
maximum number of 1/R(1,1) singularities in the basket of residual
singularities. In particular, there are no Fano polygons without
T-singularities and with a basket given by (i) {k x 1/R(1,1)} where k is a
positive integer and R>4, or (ii) {1/R1(1,1), 1/R2(1,1), 1/R3(1,1)}.
