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We show that the newly discovered BiS$_2$-based superconductors may have adominant triplet pairing component, in addition to a subdominant singletcomponent arising from the spin-orbital coupling. The pairing respectstime-reversal symmetry. The dominant triplet gap causes gap sign changesbetween the spin-split Fermi pockets. Within a pocket, the gap functionrespects $d^*_{x^2-y^2}$-wave symmetry, where the star indicates jointspin-lattice rotations. Below the Lifshitz filling level the gap is nodelss andthe superconducting state is weak topological. Above the Lifshitz points thegap becomes nodal. The dominant triplet pairing is consistent with theexperimental upper critical field exceeding the Pauli limit.

Pairing with dominant triplet component and possible weak topological superconductivity in BiS$_2$-based superconductors