  In this paper, we study Riccati solutions of Painlev\'e equations from a view
point of geometry of Okamoto-Painlev\'e pairs $(S,Y)$. After establishing the
correspondence between (rational) nodal curves on $S-Y$ and Riccati solutions,
we give the complete classification of the configurations of nodal curves on
$S-Y$ for each Okamoto-Painlev\'e pair $(S, Y)$. As an application of the
classification, we prove the non-existence of Riccati solutions of Painlev\'e
equations of types $P_{I}, P_{III}^{\tilde{D}_8}$ and $P_{III}^{\tilde{D}_7}$.
We will also give a partial answer to the conjecture in (STT) and (T) that the
dimension of the local cohomology $H^1_{Y_{red}}(S,\Theta_S(-\log Y_{red}))$ is
one.
