This paper presents an experimental and numerical investigation of the power extraction from the oscillations of a square beam due to aeroelastic instabilities. The energy harvesting is performed using a coil-magnet arrangement connected to a variable resistance load with the target objective to auto-power a remote sensor. Two aeroelastic phenomena are investigated: Vortex Induced Vibration (VIV) and cross-flow galloping. The first instability (VIV) is analyzed on a free-standing vertical structure. A second experimental set-up is developed on a horizontal square cylinder supported by springs, free to oscillate vertically as a rigid body. In this case, both galloping and VIV interact, leading to interesting characteristics in order to harvest energy from the wind. The behavior of each electro-mechanical aeroelastic system is investigated for different reduced wind speeds and load resistances in a wind tunnel. Observed efficiencies are rather low, but large enough to power a remote sensor with an adapted measuring strategy. Both harvesting systems are then studied numerically using a wake oscillator model (for VIV) coupled to a quasi-steady model (for galloping) and an electric model (for the harvester). This mathematical model is used to extend the parametric space and to highlight the effectiveness of the high stable branch of the VIV-galloping curve to harvest energy.