An inessential real singularity of a real algebraic variety is defined to be a singular point with the complex link a rational homology sphere. For such a singularity, there
is a canonical quadratic form defined on the top homology of the real link of the singularity. In the present work, this form is studied for inessential singularities of real
algebraic surfaces. A method is presented to calculate the canonical quadratic form in terms of the very good resolution graph of a real unibranch surface singularity
for which the value of the form is the Thurston-Bennequin number of the real link of the singularity. As a special case of unibranch surface singularities, the behaviour
of the canonical quadratic form is investigated on the Brieskorn double points. Furthermore, it is shown that the Thurston-Bennequin number of a real unibranch
singularity of a suspension over C^2 naturally splits into a pair of finer invariants which can be combined in a relative Thurston-Bennequin form.

Keywords : real algebraic surface, singularity, resolution graph, contact manifold, Thurston-Bennequin invariant