The spin dynamics in [TbPc2](0) and [DyPc2](0) single-molecule magnets have been investigated by means of muon and nuclear spin-lattice relaxation rate measurements. The correlation time for the spin fluctuations was found to be close to 0.1 ms already at 50 K, about 2 orders of magnitude larger than the one previously found in other lanthanide-based single-molecule magnets. In [TbPc2](0) two different regimes for the spin fluctuations have been evidenced: a high-temperature activated one involving spin fluctuations across a barrier Delta similar or equal to 880 K separating the ground and first excited states and a low-temperature regime involving quantum fluctuations within the twofold degenerate ground state. In [DyPc2](0) a high-temperature activated spin dynamics is also evidenced which, however, cannot be explained in terms of a single spin-phonon coupling constant.