Abstract: We study theoretically the phonon-induced relaxation and decoherence times of singlet-triplet qubits in lateral double quantum dots (DQDs) in GaAs/AlGaAs and Si/SiGe heterostructures. We show that the two-phonon processes can dominate over the one-phonon processes even at rather low temperature, e.g. for the decoherence time at 50 mK for biased GaAs/AlGaAs DQDs. We find that when the DQD is unbiased the qubit lifetimes are orders of magnitude longer than for the case of biased DQD. The lifetimes of the qubits in Si/SiGe DQDs are remarkably longer than the ones in GaAs/AlGaAs DQDs, and following interest in S-T_- qubits we consider the phonon-induced decoherence and relaxation particularly in the region of the singlet-triplet anticrossing. We show that the hybridization of the singlet state plays a crucial role. Among other parameters of the system, we study the effect of micromagnet-induced magnetic field gradient on the qubit, which reveals an non-trivial behavior of the relaxation and decoherence times as a result of interplay between one-phonon and two-phonon processes. We suggest how our theory can be tested experimentally.