Applications such as optical clocks, gas spectroscopy, and optical coherent tomography stand to benefit significantly from the integration of on-chip supercontinuum sources. Materials exhibiting both second- and third-order nonlinearities are particularly attractive for this purpose, enabling the implementation of f-to-2f interferometry within a single device. Aluminum nitride is an especially promising material, featuring a wide band gap of 6.2 eV and transparency down to 200 nm. Through advancements in AlN epilayer quality achieved via metalorganic vapor-phase epitaxy on sapphire and the optimization of waveguide fabrication process, we drastically reduced absorption and scattering losses compared to their polycrystalline AlN counterparts. This achievement translates into propagation losses below 1 dB/cm at 1550 nm. Moreover, by tailoring the dispersion characteristics of the waveguide, we can precisely control the positions of short- and long-wavelength dispersive waves. This enabled the generation of gap-free supercontinuum spanning from visible to mid-infrared wavelengths, by pumping with telecom femtosecond laser. When pumped with TM polarization, efficient second-harmonic generation was achieved with the phase-matched higher-order mode TM20. For specific waveguide cross sections, the SH generated component overlapped with the dispersive wave, creating ideal conditions for f-to-2f interference.