Frankly, there aren't many calculations out there. Most of the work I know of is on the calculation of the $RO(G)$-graded cohomology of a point, of a projective space, or of $B_GO(n)$. Here are some references and notes on them:

[1] L. G. Lewis, Jr., The $RO(G)$-graded equivariant ordinary cohomology of complex projective spaces with linear $\mathbb{Z}/p$ actions, in Algebraic topology and transformation groups, Lecture Notes in Math. v. 1361, 1988. (MR 979507)

In an appendix, this has the first published account of Stong's calculation of the $RO(G)$-graded cohomology of a point for $G= \mathbb{Z}/p$, where $p$ is prime. Stong calculated the multiplicative structure for $p = 2$ and $3$, and Lewis extended that to all primes. The coefficient system used is the Burnside ring system, which evaluates to the Burnside ring $A(H)$ at the orbit $G/H$.

The main purpose of the paper, though is the calculation of the cohomology of complex projective spaces, that is, the spaces $\mathbb{C}P(V)$ where $V$ is a finite- or countably infinite-dimensional complex representation of $G$. The calculation includes the multiplicative structure.

[2] W. C. Kronholm, The $RO(G)$-graded Serre spectral sequence, Homology, Homotopy Appl. 12 (2010), pp. 75-92. (MR 2607411)

Kronolm gives a similar calculation for real projective spaces, for $G=\mathbb{Z}/2$ and with constant $\mathbb{Z}/2$ coefficients.

[3] D. Dugger, Bigraded cohomology of $\mathbb{Z}/2$-equivariant Grassmannians, Geom. Topol. 19 (2015), pp.113-170. (MR 2240234)

Dugger uses Kronholm's result to calculate the $RO(G)$-graded cohomology of $B_GO(n)$ for $G = \mathbb{Z}/2$ and constant $\mathbb{Z}/2$ coefficients.

There are a couple of other papers out there with partial calculations of the cohomology of a point, but for solid calculations of the cohomology of interesting spaces, this is all I've got.

Frankly, there aren't many calculations out there. Most of the work I know of is on the calculation of the $RO(G)$-graded cohomology of a point, of a projective space, or of $B_GO(n)$. Here are some references and notes on them:

[1] L. G. Lewis, Jr., The $RO(G)$-graded equivariant ordinary cohomology of complex projective spaces with linear $\mathbb{Z}/p$ actions, in Algebraic topology and transformation groups, Lecture Notes in Math. v. 1361, 1988. (MR 979507)

In an appendix, this has the first published account of Stong's calculation of the $RO(G)$-graded cohomology of a point for $G= \mathbb{Z}/p$, where $p$ is prime. Stong calculated the multiplicative structure for $p = 2$ and $3$, and Lewis extended that to all primes. The coefficient system used is the Burnside ring system, which evaluates to the Burnside ring $A(H)$ at the orbit $G/H$.

The main purpose of the paper, though is the calculation of the cohomology of complex projective spaces, that is, the spaces $\mathbb{C}P(V)$ where $V$ is a finite- or countably infinite-dimensional complex representation of $G$. The calculation includes the multiplicative structure.

[2] W. C. Kronholm, The $RO(G)$-graded Serre spectral sequence, Homology, Homotopy Appl. 12 (2010), pp. 75-92. (MR 2607411)

Kronholm gives a similar calculation for real projective spaces, for $G=\mathbb{Z}/2$ and with constant $\mathbb{Z}/2$ coefficients.

[3] D. Dugger, Bigraded cohomology of $\mathbb{Z}/2$-equivariant Grassmannians, Geom. Topol. 19 (2015), pp.113-170. (MR 2240234)

Dugger uses Kronholm's result to calculate the $RO(G)$-graded cohomology of $B_GO(n)$ for $G = \mathbb{Z}/2$ and constant $\mathbb{Z}/2$ coefficients.

There are a couple of other papers out there with partial calculations of the cohomology of a point, but for solid calculations of the cohomology of interesting spaces, this is all I've got.