An extensive investigation of quantum oscillations in ultrasonic attenuation in white tin has been carried out with pulsed longitudinal sound waves of frequencies 390 and 430 MHz in magnetic fields up to 88 kG. The data were recorded digitally and analysed by means of the Fourier transform on a computer. Comparison of nearly forty of the observed frequency branches with previous data collected by de Haas-van Alphen and RF size effect produced, in general, excellent agreement. Several new branches confirm the observation of Matthey et al. (1974) that Craven's model of the Fermi surface of white tin needs modification in the fourth and sixth zones. Nine frequency branches (detected when the magnetic field was nearly parallel to (001)) have been assigned to magnetic breakdown orbits. In addition, four breakdown orbits were observed with the field direction near (110), two of which constitute the first observation of tunnelling at the ends of the third zone dumbbell. A comparison of the theoretical and measured oscillation amplitudes of the breakdown orbits in the (110) plane yields the breakdown fields H₀^A approximately 50 kG at the dumbbell waist and H_O^B approximately 420 kG at the ends.