(Abridged) The structural parameters of a magnitude-limited sample of 148 unbarred S0-Sb galaxies were analyzed to derive the intrinsic shape of their bulges. We developed a new method to derive the intrinsic shape of bulges based on the geometrical relationships between the apparent and intrinsic shapes of bulges and disks. The equatorial ellipticity and intrinsic flattening of bulges were obtained from the length of the apparent major and minor semi-axes of the bulge, twist angle between the apparent major axis of the bulge and the galaxy line of nodes, and galaxy inclination. We found that the intrinsic shape is well constrained for a subsample of 115 bulges with favorable viewing angles. A large fraction of them is characterized by an elliptical section (B/A<0.9). This fraction is 33%, 55%, and 43% if using their maximum, mean, or median equatorial ellipticity, respectively. Most are flattened along their polar axis (C<(A+B)/2). The distribution of triaxiality is strongly bimodal. This bimodality is driven by bulges with Sersic index n>2, or equivalently, by the bulges of galaxies with a bulge-to-total ratio B/T>0.3. In particular, bulges with nleq2 and with B/Tleq0.3 show a larger fraction of oblate axisymmetric (or nearly axisymmetric) bulges, a smaller fraction of triaxial bulges, and fewer prolate axisymmetric (or nearly axisymmetric) bulges with respect to bulges with n>2 and with B/T>0.3, respectively. According to predictions of the numerical simulations of bulge formation, bulges with nleq2, which show a high fraction of oblate axisymmetric (or nearly axisymmetric) shapes and have B/Tleq0.3, could be the result of dissipational minor mergers. Both major dissipational and dissipationless mergers seem to be required to explain the variety of shapes found for bulges with n>2 and B/T>0.3.
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