  We show that the number of lines contained in a supersingular quartic surface
is 40 or at most 32, if the characteristic of the field equals 2, and it is
112, 58, or at most 52, if the characteristic equals 3. If the quartic is not
supersingular, the number of lines is at most 60 in both cases. We also give a
complete classification of large configurations of lines.
