The real question is:
What do I gain from spin contamination?
Spin contamination is usually presented as a bad thing. However, the extra (sometimes large) spin polarization that can occur when relaxing the spatial constraint on the wavefunction is what gives UHF a little bit of electron correlation beyond RHF and ROHF. This is why you should be wary of people who blindly state "there is no correlation in Hartree-Fock". They probably mean "when compared to wavefunction-based methods", but without context, it isn't true!
I will steal from Frank Jensen's book, section 4.4:
By including electron correlation in the wave function, the UHF method introduces more biradical character into the wave function than RHF. The spin contamination part is also purely biradical in nature, i.e. a UHF treatment in general will overestimate the biradical character of the wave function. Most singlet states are well described by a closed shell wave function near the equilibrium geometry and, in those cases, it is not possible to generate a UHF solution that has a lower energy than the RHF. There are systems, however, for which this does not hold. An example is the ozone molecule, where two types of resonance structures can be drawn.
The biradical resonance structure for ozone requires two singly occupied MOs, and it is clear that an RHF type wave function, which requires all orbitals to be doubly occupied, cannot describe this. A UHF type wave function, however, allows the $\alpha$ and $\beta$ orbitals to be spatially different, and can to a certain extent incorporate both resonance structures. Systems with biradical character will in general have a (singlet) UHF wave function different from an RHF.
If you look for the singlet with RHF, there will be almost no biradical character in the resulting wavefunction, unless there is a biradical-looking orbital delocalized over the whole molecule. I don't know if there is, just guessing. With the biradical you can also see how the triplet state will enter the picture; I am leaving the details to Jensen's chapter. There is an analogy between UHF including too much biradical character with MP2 overestimating the dispersion interaction; it is the first level of theory where the effect is being considered qualitatively, and should not be expected to be quantitatively accurate. Though, UHF is probably worse at describing biradical character than MP2 is at quantifying the dispersion energy.
I haven't thought much about if this is considered to be dynamic or static correlation if we perform that artificial division. For example, this singlet biradical can be described by two determinants composing a single configuration state function (so TCSCF would work, and some flavors of ROHF), but you can capture the same effect after the fact with EOM-CCSD. It might be considered a static effect, since the presence of higher spin states in the resulting wavefunction results in some configurational mixing.