  We provide explicit graded constructions of orbifold del Pezzo surfaces with
rigid orbifold points of type $\left\{k_i\times\frac{1}{r_i}(1,a_i): 3\le r_i
\le 10,k_i \in \ZZ_{\ge 0}\right\}$; as well-formed and quasismooth varieties
embedded in some weighted projective space. In particular, we present a
collection of 147 such surfaces such that their image under their
anti-canonical embeddings can be described by using one of the following sets
of equations: a single equation, two linearly independent equations, five
maximal Pfaffians of $5\times 5$ skew symmetric matrix, and nine $2\times 2$
minors of size 3 square matrix. This is a complete classification of such
surfaces under certain carefully chosen bounds on the weights of ambient
weighted projective spaces and it is largely based on detailed
computer-assisted searches by using the computer algebra system \textsc{magma}.
