As always, it helps to RTFM (read the manual). Linn has an excellent one for the LP12. In particular it stresses that the LP12 should stand on a lightweight surface, such as a light coffee table. If wall mounted, the shelf should be low mass.
The Linn is a low mass design with long-travel springs. Its resonant frequency is very low, but if teemed with a massive stand which itself absorbs energy and resonates around the same frequency, the two resonances can reinforce.