  We study the shape coexistence in the nucleus $^{28}$Si with the nuclear
shell model using numerical diagonalizations complemented with variational
calculations based on the projected generator-coordinate method. The
theoretical electric quadrupole moments and transitions as well as the
collective wavefunctions indicate that the standard USDB interaction in the
$sd$ shell describes well the ground-state oblate rotational band, but misses
the experimental prolate band. Guided by the quasi-SU(3) model, we show that
the prolate band can be reproduced in the $sd$ shell by reducing the energy of
the $0d_{3/2}$ orbital. Alternatively, in the extended $sdpf$ configuration
space the SDPF-NR interaction, which describes well other Si isotopes, also
reproduces the oblate and prolate bands. Finally, we address the possibility of
superdeformation in $^{28}$Si within the $sdpf$ space. Our results disfavour
the appearance of superdeformed states with excitation energy below 20 MeV.
