A size-able amount of time and energy has been spent by the "hard condensed matter" community over the past quarter century studying unconventional superconductors. A nice and recent review is by Mike Norman. In the absence of spin-orbit coupling spin is a good quantum number and the Cooper pairs must either be in a spin singlet or a spin triplet state. Furthermore, in a crystal with inversion symmetry spin singlets (triplets) are associated with even (odd) parity.
Actually, pinning down the symmetry of the Cooper pairs from experiment turns out to be extremely tricky. In the cuprates the "smoking gun" experiments that showed they were really d-wave used cleverly constructed Josephson junctions, that allowed one to detect the phase of the order parameter and show that it changed sign as one moved around the Fermi surface.
How can one show that the pairing is spin triplet?
Perhaps the simplest way is to show that they have an upper critical magnetic field that exceeds the Clogston-Chandrasekhar limit [often called the Pauli paramagnetic limit, but I think this is a misnomer].
In most type II superconductors the upper critical field is determined by orbital effects. When the magnetic field gets large enough the spacing between the vortices in the Abrikosov lattice becomes comparable to the size of the vortices [determined by the superconducting coherence length in the directions perpendicular to the magnetic field direction]. This destroys the superconductivity. In a layered material the orbital upper critical field can become very large for fields parallel to the layers, because the interlayer coherence length can be of the order of the lattice spacing. Consequently, the superconductivity can be destroyed by the Zeeman effect breaking up the singlets of the Cooper pairs. This is the Clogston-Chandrasekhar paramagnetic limit.
How big is this magnetic field?
In a singlet superconductor the energy lost compared to the metallic state is ~chi_s B^2/2 where chi_s is the magnetic [Pauli spin] susceptibility in the metallic phase. Once the magnetic field is large enough that this is larger than the superconducting condensation energy, superconductivity becomes unstable. Normally, these two quantities are compared within BCS theory, and one finds that the "Pauli limit", H_P = 1.8 k_B T_c/g mu_B. This means for a Tc=10K the upper critical field is 18 Tesla.
Sometimes, people then use this criteria to claim evidence for spin triplets.
However, in 1999 I realised that one could estimate the upper critical field independent of BCS theory, using just the measured values of the spin susceptibility and condensation energy. In this PRB my collaborators and I showed that for the organic charge transfer salt kappa-(BEDT-TTF)2Cu(SCN)2 the observed upper critical field of 30 Tesla agreed with the theory-independent estimate. In contrast, the BCS estimate was 18 Tesla. Thus, the experiment was consistent with singlet superconductivity.
But, exceeding the Clogston-Chandrasekhar paramagnetic limit is the first hint that one might have a triplet superconductor. Indeed this was the case for the heavy fermion superconductor UPt3, but not for Sr2RuO4. Recent, phase sensitive Josephson junction measurements have shown that both these materials have odd-parity superconductivity, consistent with triplet pairing. A recent review considered the status of the evidence for triplet odd-parity pairing and the possibility of a topological superconductor in Sr2RuO4.
A PRL from Nigel Hussey's group last year reported measurements of the upper critical field for the quasi-one dimensional material Li0.9Mo6O17. They found that the relevant upper critical field was 8 Telsa, compared to values of 5 Tesla and 4 Tesla for the thermodynamic and BCS estimates, respectively.
Thus, this material could be a triplet superconductor.
[Aside: this material has earlier attracted considerable attention because it has a very strange metallic phase, as reviewed here.]
A challenge is to now come up with more definitive experimental signatures of the unconventional superconductivity. Given the history of UPt3 and Sr2RuO4, this could be a long road... but a rich journey....