The exploitation of the two-dimensional electron gas (2DEG) channel of an AlN/GaN/AlN double barrier heterostructure, for High Electron Mobility Transistors (HEMTs) with metal-oxidesemiconductor (MOS) tri-gate around a fin-shaped channel (MOS-FinHEMTs), has been investigated by combining fabrication, dc I-V measurements and simulations of single-fin MOS-FinHEMT devices. The dependence of the threshold voltage (V-th) and the maximum drain-source current (I-ds,I-max) on the fin width (W-fin), as well as the effects of ohmic contact resistance, gate-drain and source-gate distance and of the Al2O3 gate dielectric thickness (t(ox)), have been addressed. Fabricated single-fin MOS-FinHEMT devices with t(ox) = 20 nm, exhibited a positive shift of V-th, in comparison to a reference planar-gate device, ranging from +0.8 V for W-fin = 650 nm to + 3.4 V for W-fin = 200 nm, due to lateral depletion of the channel by the gate contacts on the fin sidewalls. Simulations reproduced the experimental Vth values and also predicted the Vth of devices with narrower fins, down to W-fin = 10 nm. The boundary for normally-off operation (V-th = 0 V) was determined at W-fin = 17 nm that may increase up to 31 nm if the tensile strain of the top AlN barrier in the fin nanostructure is elastically relaxed. A reduction of maximum drain-source current per top gate width (I-ds,I-max/W-g) with decreasing W-fin in the range of 200-650 nm may result from increased ohmic contact resistance. However, for narrower fins, I-ds,I-max/W-g is predicted to decrease significantly with decreasing W-fin, due to the lateral field of the sidewall gates. The I-ds,I-max/W-g will also decrease with increasing distance between the source, gate and drain contacts for any W-fin. Finally, the V-th and I-ds,I-max/W-g values were calculated for Al2O3 thickness in the range of 5 to 40 nm.