In recent years, the possibility of half-quantum vortices (HQVs) in the ruthenate superconductor Sr₂RuO₄ (SRO) has generated great interest. Experimental evidence suggests that SRO is a spin-triplet odd-parity superconductor analogous to the A-phase of superfluid ³He. Importantly, superconductivity of this type may be capable of supporting remarkable excitations, such as vortices that carry half a unit of magnetic flux, and may be used to supply topologically-protected states for quantum computation. HQVs have not been observed in large samples, but they may be more stable, and thus observable, in mesoscopic samples [1]. In this talk, I will present a new method for ultrasensitive cantilever magnetometry that allows us to probe the magnetic response of mesoscopic samples of SRO. Using this technique, we have detected the entry of individual vortices into micron-size rings of SRO. Our most intriguing observation is the appearance of fractional fluxoid states that have half the magnetic moment of the full (integer) fluxoid. We find that the stability region of the half-fluxoid state grows linearly with the magnitude of the magnet-ic field applied perpendicular to the crystal c-axis. Although this finding does not directly prove the existence of HQVs in this system, we find that the observed field dependence of the stability region of the half-fluxoid state is consistent with the existence of a spontaneous spin polarization predicted to exist in the HQV state [2].