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First, when $n>0$, there is the Penner decomposition (see e.g. Harer, The cohomology of moduli spaces, LNM 1337 or Penner, Comm Math Phys 113, 299-339). This gives in principle a finite-dimensional complex that computes the cohomology of the coarse moduli spaces. In practice however the number of cells quickly gets quite large and I'm not sure whether or not someone has written a program that implements this.

When $g=0$, the cohomology was computed by S. Keel (Transactions AMS, 330, 2, 545-574) for any number of points. Keel also computes the action of the symmetric groups, the cup product and the rational mixed Hodge structure, which turns out to be a direct sum of Tate ones.

[upd: Keel is primarily interested in the cohomology of the Deligne-Mumford compactification and it is not clear to me how to deduce the cohomology of the open part from his results. So here is an ad hoc way to describe the cohomology of $M_{0,n}$. Let $H=H^*(\mathbf{P}^1(\mathbf{C}),\mathbf{Q})$. Following B. Totaro (Configuration spaces of algebraic varieties, Topology 35 (1996), no. 4, 1057--1067) form the cdg-algebra $$F=H^{\otimes n}[a_{i,j}]/rels$$ where $a_{i,j},i,j=1,\ldots, n, i\neq j$ are variables in degree 1 (so they anti-commute with everything) and the relations $rels$ are

First, when $n>0$, there is the Penner decomposition (see e.g. Harer, The cohomology of the moduli space of curves, LNM 1337 or Penner, Comm Math Phys 113, 299-339). This gives in principle a finite-dimensional complex that computes the cohomology of the coarse moduli spaces. In practice however the number of cells quickly gets quite large and I'm not sure whether or not someone has written a program that implements this.

When $g=0$, the cohomology was computed by S. Keel (Transactions AMS, 330, 2, 545-574) for any number of points. Keel also computes the action of the symmetric groups, the cup product and the rational mixed Hodge structure, which turns out to be a direct sum of Tate ones.

[upd: Keel is primarily interested in the cohomology of the Deligne-Mumford compactification and it is not clear to me how to deduce the cohomology of the open part from his results. So here is an ad hoc way to describe the cohomology of $M_{0,n}$. Let $H=H^*(\mathbf{P}^1(\mathbf{C}),\mathbf{Q})$. Following B. Totaro (Configuration spaces of algebraic varieties, Topology 35 (1996), no. 4, 1057--1067) form the cdg-algebra $$F=H^{\otimes n}[a_{i,j}]/rels$$ where $a_{i,j},i,j=1,\ldots, n, i\neq j$ are variables in degree 1 (so they anti-commute with everything) and the relations $rels$ are

Let me also mention two results on the Euler characteristics that extend Harer-Zagier. Bini and Harer give an explicit formula for the Euler characteristic of the Deligne-Mumford compactified moduli spaces in http://arxiv.org/abs/math/0506083; E. Gorsky http://arxiv.org/abs/0906.0841 computes the $S_n$-equivariant Euler characteristic of $M_{g,n}$ for an abritrary $n$. By Getzler-Kapranov this also gives the equivariant Euler characteristic of the compactified moduli spaces.

Let me also mention two results on the Euler characteristics that extend Harer-Zagier. Bini and Harer give an explicit formula for the Euler characteristic of the Deligne-Mumford compactified moduli spaces in https://arxiv.org/abs/math/0506083; E. Gorsky https://arxiv.org/abs/0906.0841 computes the $S_n$-equivariant Euler characteristic of $M_{g,n}$ for an arbitrary $n$. By Getzler-Kapranov this also gives the equivariant Euler characteristic of the compactified moduli spaces.

Complementing other answers in this thread:

First, when $n>0$, there is the Penner decomposition (see e.g. Harer, The cohomology of moduli spaces, LNM 1337 or Penner, Comm Math Phys 113, 299-339). This gives in principle a finite-dimensional complex that computes the cohomology of the coarse moduli spaces. In practice however the number of cells quickly gets quite large and I'm not sure whether or not someone has written a program that implements this.

When $g=0$, the cohomology was computed by S. Keel (Transactions AMS, 330, 2, 545-574) for any number of points. Keel also computes the action of the symmetric groups, the cup product and the rational mixed Hodge structure, which turns out to be a direct sum of Tate ones.

[upd: Keel is primarily interested in the cohomology of the Deligne-Mumford compactification and it is not clear to me how to deduce the cohomology of the open part from his results. So here is an ad hoc way to describe the cohomology of $M_{0,n}$. Let $H=H^*(\mathbf{P}^1(\mathbf{C}),\mathbf{Q})$. Following B. Totaro (Configuration spaces of algebraic varieties, Topology 35 (1996), no. 4, 1057--1067) form the cdg-algebra $$F=H^{\otimes n}[a_{i,j}]/rels$$ where $a_{i,j},i,j=1,\ldots, n, i\neq j$ are variables in degree 1 (so they anti-commute with everything) and the relations $rels$ are

1. $a_{i,j}=a_{j,i}$;

2. the cyclic permutations of $a_{i,j}a_{j,k}$ add up to 0 where $i,j,k$ are pairwise distinct;

3. $a_{i,j}(h_i-h_j)=0$ where $h_i=1\otimes\ldots\otimes h\otimes\ldots\otimes 1$ ($h\in H$ in the $i$-th place), and similarly for $h_j$.

The differential annihilates $H^{\otimes n}$ and takes $a_{i,j}$ to the pullback of the class of the diagonal under the projection $p_{i,j}:\mathbf{P}^1(\mathbf{C})^{\times n}\to \mathbf{P}^1(\mathbf{C})^{\times 2}$ to the $i$-th and $j$-th factors.

The cohomology of $F$ is the cohomology of the space $F(\mathbf{P}^1(\mathbf{C}),n)$ of ordered $n$-tuples of distinct points in $\mathbf{P}^1(\mathbf{C})$. The moduli space is the quotient of $F(\mathbf{P}^1(\mathbf{C}),n)$ by the action of $PGL_2(\mathbf{C})$. Now for this action the Leray-Hirsch principle holds: there is a degree 3 class whose restriction to each orbit generates the $H^3$ of the orbit. To see this note that $F(\mathbf{P}^1(\mathbf{C}),3)\cong PGL_2(\mathbf{C})$. If we take a generator of $H^3( F(\mathbf{P}^1(\mathbf{C}),3),\mathbf{Q})$ and take the sum of its pullbacks under all possible projections $F(\mathbf{P}^1(\mathbf{C}),n)\to F(\mathbf{P}^1(\mathbf{C}),3)$, this should do the trick. So $$H^{\ast}(F(\mathbf{P}^1(\mathbf{C}),n))\cong H^{\ast}(PGL_2(\mathbf{C}))\otimes H^{\ast}(M_{0,n})$$ with $\mathbf{Q}$-coefficients.

So the recipe to compute say the complex the cohomology of $M_{0,n}$ as an $S_n$-module is as follows: form a polynomial $f(t)=\sum c_i t^i$ where $c_i=H^i(F)$ viewed as an element of the representation ring $R(S_n)$ of $S_n$. This polynomial is the product of $1+t^3$ and some other polynomial $g$, which will be the $S_n$-equivariant Poincar'e polynomial of $M_{0,n}$.

Note also that if one is interested only in the Poincar'e polynomial and not in the action of $S_n$, then the answer is simply $(1+2t)(1+3t)\cdots (1+(n-1)t)$.

I'm not sure though what the reference for this is or whether there is a better description of $H^*(M_{0,n},\mathbf{Q})$ or whether the answer has been tabulated for small $n$. I'd be interested to know the answer to either of these questions.]

Let me also mention two results on the Euler characteristics that extend Harer-Zagier. Bini and Harer give an explicit formula for the Euler characteristic of the Deligne-Mumford compactified moduli spaces in http://arxiv.org/abs/math/0506083; E. Gorsky http://arxiv.org/abs/0906.0841 computes the $S_n$-equivariant Euler characteristic of $M_{g,n}$ for an abritrary $n$. By Getzler-Kapranov this also gives the equivariant Euler characteristic of the compactified moduli spaces.

Complementing other answers in this thread:

First, when $n>0$, there is the Penner decomposition (see e.g. Harer, The cohomology of moduli spaces, LNM 1337 or Penner, Comm Math Phys 113, 299-339). This gives in principle a finite-dimensional complex that computes the cohomology of the coarse moduli spaces. In practice however the number of cells quickly gets quite large and I'm not sure whether or not someone has written a program that implements this.

When $g=0$, the cohomology was computed by S. Keel (Transactions AMS, 330, 2, 545-574) for any number of points. Keel also computes the action of the symmetric groups, the cup product and the rational mixed Hodge structure, which turns out to be a direct sum of Tate ones.

[upd: Keel is primarily interested in the cohomology of the Deligne-Mumford compactification and it is not clear to me how to deduce the cohomology of the open part from his results. So here is an ad hoc way to describe the cohomology of $M_{0,n}$. Let $H=H^*(\mathbf{P}^1(\mathbf{C}),\mathbf{Q})$. Following B. Totaro (Configuration spaces of algebraic varieties, Topology 35 (1996), no. 4, 1057--1067) form the cdg-algebra $$F=H^{\otimes n}[a_{i,j}]/rels$$ where $a_{i,j},i,j=1,\ldots, n, i\neq j$ are variables in degree 1 (so they anti-commute with everything) and the relations $rels$ are

1. $a_{i,j}=a_{j,i}$;

2. the cyclic permutations of $a_{i,j}a_{j,k}$ add up to 0 where $i,j,k$ are pairwise distinct;

3. $a_{i,j}(h_i-h_j)=0$ where $h_i=1\otimes\ldots\otimes h\otimes\ldots\otimes 1$ ($h\in H$ in the $i$-th place), and similarly for $h_j$.

The differential annihilates $H^{\otimes n}$ and takes $a_{i,j}$ to the pullback of the class of the diagonal under the projection $p_{i,j}:\mathbf{P}^1(\mathbf{C})^{\times n}\to \mathbf{P}^1(\mathbf{C})^{\times 2}$ to the $i$-th and $j$-th factors.

The cohomology of $F$ is the cohomology of the space $F(\mathbf{P}^1(\mathbf{C}),n)$ of ordered $n$-tuples of distinct points in $\mathbf{P}^1(\mathbf{C})$. The moduli space is the quotient of $F(\mathbf{P}^1(\mathbf{C}),n)$ by the action of $PGL_2(\mathbf{C})$. Now for this action the Leray-Hirsch principle holds: there is a degree 3 class whose restriction to each orbit generates the $H^3$ of the orbit. To see this note that $F(\mathbf{P}^1(\mathbf{C}),3)\cong PGL_2(\mathbf{C})$. If we take a generator of $H^3( F(\mathbf{P}^1(\mathbf{C}),3),\mathbf{Q})$ and take the sum of its pullbacks under all possible projections $F(\mathbf{P}^1(\mathbf{C}),n)\to F(\mathbf{P}^1(\mathbf{C}),3)$, this should do the trick. So $$H^{\ast}(F(\mathbf{P}^1(\mathbf{C}),n))\cong H^{\ast}(PGL_2(\mathbf{C}))\otimes H^{\ast}(M_{0,n})$$ with $\mathbf{Q}$-coefficients.

So the recipe to compute say the complex the cohomology of $M_{0,n}$ as an $S_n$-module is as follows: form a polynomial $f(t)=\sum c_i t^i$ where $c_i=H^i(F)$ viewed as an element of the representation ring $R(S_n)$ of $S_n$. This polynomial is the product of $1+t^3$ and some other polynomial $g$, which will be the $S_n$-equivariant Poincar'e polynomial of $M_{0,n}$.

Note also that if one is interested only in the Poincar'e polynomial and not in the action of $S_n$, then the answer is simply $(1+2t)(1+3t)\cdots (1+(n-2)t)$.

I'm not sure though what the reference for this is or whether there is a better description of $H^*(M_{0,n},\mathbf{Q})$ or whether the answer has been tabulated for small $n$. I'd be interested to know the answer to either of these questions.]

Let me also mention two results on the Euler characteristics that extend Harer-Zagier. Bini and Harer give an explicit formula for the Euler characteristic of the Deligne-Mumford compactified moduli spaces in http://arxiv.org/abs/math/0506083; E. Gorsky http://arxiv.org/abs/0906.0841 computes the $S_n$-equivariant Euler characteristic of $M_{g,n}$ for an abritrary $n$. By Getzler-Kapranov this also gives the equivariant Euler characteristic of the compactified moduli spaces.