ATLAS has a fairly new result out measuring the properties of the Higgs boson using leptonic and diphoton final states. In particular, they compare the expected results for a Standard Model JP = 0+ Higgs to several alternative models. The measurement pretty clearly rules out things like a spin-2 Higgs and also sets upper bounds on the strengths of beyond the Standard Model Higgs tensor couplings. The spin-parity state can often be determined from looking at final state kinematic distributions. Things like the angular distribution will often be very different for different types of particles.
This kind of result is nice because it helps bolster the case that the ~125 GeV particle found at the LHC has the same fundamental properties as the Standard Model Higgs boson. Most other measurements are measuring cross sections or branching fractions, which tell us about the particle’s interactions with other particles. So far, everything looks remarkably like the Standard Model Higgs.