Results of extensive Cu NMR and NQR studies in the superconducting La2-xSrxCuO4 with x=0.075-0.24 have been presented. The rapid decrease of the nuclear spin-lattice relaxation rate, 1/T1, and the spin Knight shift below T(c) confirm microscopically the existence of the superconducting phase with lower T(c) than 38 K for 0.10 less-than-or-equal-to x less-than-or-equal-to 0.24. For x = 0.10, magnetic anomalies emerge at low-T in some small parts of the sample, e.g. T-independent T1 and broadening of spectrum, having some relevance to the magnetic order in La2 BaxCuO4 with x = 0.125. For x>0.15, the T1T = const. behavior and the existence of the residual spin Knight shift far below T(c) have revealed that the superconductivity is of a gapless type with a finite density of states at the Fermi level caused by the mixture of different structures, i.e. tetragonal and orthorhombic.