  We develop a theory of Sobolev orthogonal polynomials on the Sierpi\'nski
gasket ($SG$). These orthogonal polynomials arise through the Gram-Schmidt
orthogonalisation process applied on the set of monomials on $SG$ using several
notions of a Sobolev inner products. After establishing some recurrence
relations for these orthogonal polynomials, we give estimates for their $L^2$,
$L^\infty$ and Sobolev norms, and study their asymptotic behaviour. Finally, we
study the properties of zero sets of polynomials and develop fast computational
tools to explore applications to quadrature and interpolation.
