No.
SOLAS does not require any ship to be classed. However, SOLAS regulation II-1/3-1 does provide that, in addition to the requirements contained elsewhere in the SOLAS regulations, ships must be designed, constructed and maintained in compliance with the structural, mechanical and electrical requirements of a classification society recognised by the flag State Administration, or with applicable national standards of the Administration which provide an equivalent level of safety. (In practice, few, if any, Administrations have any equivalent “national standards”.)
