We consider smooth area-preserving flows (also known as locally Hamiltonian flows) on surfaces of genus g ≥ 1 and study ergodic integrals of smooth observables along the flow trajectories. We show that these integrals display a power deviation spectrum and describe the cocycles that lead the pure power behaviour, giving a new proof of results by Forni (Annals 2002) and Bufetov (Annals 2014) and generalizing them to observables which are non-zero at fixed points. This in particular completes the proof of the original formulation of the Kontsevitch-Zorich conjecture. Our proof is based on building suitable correction operators for cocycles with logarithmic singularities over a full measure set of interval exchange transformations (IETs), in the spirit of Marmi-Moussa-Yoccoz work on piecewise smooth cocycles over IETs. In the case of symmetric singularities, exploiting former work of the second author (Annals 2011), we prove a tightness result for a finite codimension class of observables. We then apply the latter result to prove the existence of ergodic infinite extensions for a full measure set of locally Hamiltonian flows with non-degenerate saddles in any genus g ≥ 2.2000 Mathematics Subject Classification. 37E35, 37A40, 37A10, 37C83. 1 More precisely, referring to the decomposition described in § 2.1.1, we call Arnold flow the restriction to a minimal component obtained by removing the center and the disk filled by periodic orbits around it (called island), which, as Arnold shows in [2], is always bounded by a saddle loop.