Photodetectors in the ultraviolet spectral range are of great interest for applications such as fluorescence-free Raman spectroscopy and non-line-of-sight optical communications. These applications require single-photon sensitivity, resulting in the use of intensified CCDs or photomultiplier tubes (PMTs) that are bulky, fragile, operate at high voltages, and/or require active cooling. Silicon carbide avalanche photodiodes (SiC APD) show promise as a compact, rugged replacement, as they exhibit low noise, durability and high gain. In this paper we report on detailed studies of p+-p-i-n SiC APDs operating in both Geiger and linear mode. The single photon detection efficiency (SPDE) of these devices was measured as a function of excess bias using a pulsed 280 nm light emitting diode source focused through a 20 µm pinhole to a spot size <12 µm, with photons per pulse characterized at < 0.5 using a UV enhanced PMT. Devices with 100 µm diameter achieved SPDE > 20% with dark count rate (DCR) of < 700 Hz in both gated and continuous operation. Comparison with linear mode operation shows that avalanche multiplication gain in these devices exceeds 5x106 under these conditions. Examination of linear mode gain vs. applied bias dependence suggests that a sluggish dependence corresponds to a poorer SPDE, which is likely associated with parasitic resistance in these devices. This resistance is consistent with the observed inverse dependence of calculated linear mode gain with increasing optical flux. A peak SPDE of 37% was measured at an excess bias of 3.9 V with a DCR of 7.3 kHz.