The wetting transition in the transverse-field spin- XY model with opposite boundary fields is studied analytically and numerically. We find that the phase diagram is complex and that the wetting transition is of three types: first, second, and fourth order. The energy gap is obtained analytically, and the magnetization profile, correlation functions, and wetting layer thickness are obtained numerically. For , a first-order phase transition occurs at , where is the continuous wetting transition point. For larger than , the continuous wetting transition occurs at , and vice versa. For , the wetting transition is second order, and commensurate and incommensurate phases occur for and , respectively. For , the wetting transition is fourth order. For this fourth-order phase transition, the third derivative of the surface magnetization oscillates and diverges near the transition point. The correlation length exponent is , and the dynamic exponent is . Thus, this fourth-order transition belongs to a new universality class. The wetting behavior is induced by asymmetric boundary fields and its finite-size scaling is discussed.