In article , Mike Lyle
writes
Gosh! All along I thought the small regrowths must be seedlings. Live
and learn: thanks to both.

Bramble seedlings could be grown by saving seed from blackberries, but
if propagation did happen to be desired, a better way would be to let
the tips of runners root into soil. They will do that anyway if you
leave them to their own devices.
--
Alan & Joan Gould - North Lincs.