  We present the design and implementation of a mechanical low-pass filter
vibration isolation used to reduce the vibrational noise in a cryogen-free
dilution refrigerator operated at 10 mK, intended for scanning probe
techniques. We discuss the design guidelines necessary to meet the competing
requirements of having a low mechanical stiffness in combination with a high
thermal conductance. We demonstrate the effectiveness of our approach by
measuring the vibrational noise levels of an ultrasoft mechanical resonator
positioned above a SQUID. Starting from a cryostat base temperature of 8 mK,
the vibration isolation can be cooled to 10.5 mK, with a cooling power of 113
$\mu$W at 100 mK. We use the low vibrations and low temperature to demonstrate
an effective cantilever temperature of less than 20 mK. This results in a force
sensitivity of less than 500 zN/$\sqrt{\mathrm{Hz}}$, and an integrated
frequency noise as low as 0.4 mHz in a 1 Hz measurement bandwidth.
