  We demonstrate direct evidence of possible surface superconductivity on
small, well-shaped MgB2 single crystals. Transport measurements in the range H
< 1.6 Hc2^c, where Hc2^c is the bulk upper critical field for the c axis, show
non-Ohmic and strongly angular dependent resistivity. Studies of the alignment
of H with selected crystal surfaces, transport and specific heat measurements
on the same crystal, and a physical sculpturing of the crystal surfaces using a
focused ion beam all support the conclusion. Similar, albeit less pronounced
results are obtained for fields in the basal plane.
