Context. Knowledge of the intrinsic shapes of galaxy components provides crucial information when constraining phenomena driving their formation and evolution. Aims: We analize the structural parameters of a magnitude-limited sample of 148 unbarred S0-Sb galaxies to derive the intrinsic shape of their bulges. Methods: We developed a new method to derive the intrinsic shapes of bulges based on geometrical relationships between the apparent and intrinsic shapes of bulges and disks. Bulges were assumed to be triaxial ellipsoids sharing the same center and polar axis of their surrounding disks. Disks were assumed to be circular, infinitesimally thin, and to lie on the equatorial plane of bulges. The equatorial ellipticity and intrinsic flattening of bulges were obtained from the length of the apparent major and minor semi-axes of the bulge, the twist angle between the apparent major axis of the bulge and the galaxy line of nodes, and the galaxy inclination. Results: We find that the intrinsic shape is well constrained for a subsample of 115 bulges with favorable viewing angles. A large fraction of them are 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). Only 18% of the observed bulges have a probability > 50% and none has a probability > 90% of being elongated along the polar axis. The distribution of triaxiality is strongly bimodal. This bimodality is driven by bulges with Sérsic index n > 2, or equivalently, by the bulges of galaxies with a bulge-to-total ratio B/T > 0.3. Bulges with n ≤ 2 and with B/T ≤ 0.3 follow a similar distribution, which differs from that of bulges with n > 2 and B/T > 0.3. In particular, bulges with n ≤ 2 and B/T ≤ 0.3 exhibit 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. No correlation is found between the intrinsic shape and either the luminosity or velocity dispersion of bulges. Conclusions: According to predictions of the numerical simulations of bulge formation, bulges with n ≤ 2, which show a high fraction of oblate axisymmetric (or nearly axisymmetric) shapes and have B/T ≤ 0.3, may 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.