Core-collapse supernovae have been supposed tobe one of the most plausible sources of gravitational waves. Based on a series of our magnetohydrodynamic core-collapsesimulations, we find that the gravitational amplitudes at core bounce canbe within the detection limits for the currently runninglaser-interferometers for a galactic supernova if the central corerotates sufficiently rapidly. This is regardless of the difference of the realisticequations of state and the possible occurrence of the QCD phasetransition near core bounce. Even if the core rotates slowly, we pointout that thegravitational waves generated from anisotropic neutrino radiationin the postbounce phase due to the standing accretion shockinstability (SASI) could be within the detection limits of the detectors in the next generation such as LCGT and the advanced LIGO for the galactic source.Since the waveforms significantly depend on the exploding scenarios,our results suggest that we can obtain the informationabout the long-veiled explosion mechanism from the gravitational wave signals.Furthermore we discuss the gravitational wave background (GWB) from theexplosions of Pop III stars and show that the GWB from Pop III,depending on their formation rates, can be large enough to be within the detection limitsof future planned interferometers such asDECIGO and BBO in the frequency interval of ~0.1-1 Hz.This means that the detections of GW background from Pop III stars canbe an important tool to supplythe information about the star formation history in the early universe.