IS DMSO-o a σ-donor, a π-donor, or both?
The first thing to remember is that a &#x3c3;-symmetric bond is always better than a &#x3c0;-symmetric one. Orbitals that can bond with each other are typically oriented in one direction or another (except for s-orbitals, but those can only participate in &#x3c3; bonds, anyway). Approaching these orbitals from the direction ‘their lobes point towards’ and in the direction ‘your own lobe is pointing towards’ results in a much better overlap than if you attempted to create a &#x3c0; type interaction. Thus, typically any &#x3c3; interaction will be preferable to any &#x3c0; interaction between a given set of two bonding partners.

To be perfectly honest, I am not aware of any ligand which does not initially bond as a &#x3c3; donor.

The question remains what will happen if we add &#x3c0; effects into the picture. &#x3c0; effects can either be of the &#x3c0;-Lewis acidic type (e.g. $\ce{CO}$) or of the &#x3c0;-Lewis basic type (e.g. $\ce{F-}$). In the case of DMSO, the LUMO has much greater contributions on sulfur than on oxygen — in line with the general expectation that the LUMO is centred on electropositive partners while HOMOs centre more around the electronegative partners.

Also in line with what oxygen does in other ligands, I would expect DMSO-&#x3ba;O to be a &#x3c0; base, due to the probability of another populated orbital having significant contributions on oxygen. Contrarily, the picture of the LUMO shows how well sulfur in DMSO-&#x3ba;S can act as a Lewis acid.

For Lewis acidic bonding, i.e. &#x3c0; backbonding (also written as M$\leftrightarrows$L), it is beneficial if there is significant charge density on the central metal, i.e. if the metal has a somewhat Lewis basic character. Often, this will be enhanced by the presence of electrons in d-orbitals of the $\mathrm{t_{2g}}$ type.

For Lewis basic ligand bonding, i.e. &#x3c0; forward bonding (also written as L$\rightrightarrows$M), it is beneficial for the metal to be in a Lewis acidic, electron-poor state — often with empty d-orbitals.

In the DMSO-&#x3ba;S context, remember that sulfur carries a partial positive charge and is thus already electron deficient somewhat ($\pm 0$ oxidation state, but it almost has the same electronegativity as carbon, so it is not far away from $\mathrm{+II}$). Therefore, the &#x3c3; forward bond will be weak per se. Only if sufficient stabilisation through backbonding can be supplied will a DMSO-&#x3ba;S complex be sufficiently stable.