  We study here the precession of the spin of a test gyroscope attached to a
stationary observer in the Kerr spacetime, specifically, to distinguish naked
singularity (NS) from black hole (BH). It was shown recently that for gyros
attached to static observers, their precession frequency became arbitrarily
large in the limit of approach to the ergosurface. For gyros attached to
stationary observers that move with non-zero angular velocity $\Omega$, this
divergence at the ergosurface can be avoided. Specifically, for such gyros, the
precession frequencies diverge on the event horizon of a BH, but are finite and
regular for NS everywhere except at the singularity itself. Therefore a genuine
detection of the event horizon becomes possible in this case. We also show that
for a near-extremal NS ($1<a_* < 1.1$), characteristic features appear in the
radial profiles of the precession frequency, using which we can further
distinguish a near-extremal NS from a BH, or even from NS with larger angular
momentum. We then investigate the Lense-Thirring (LT) precession or nodal plane
precession frequency of the accretion disk around a BH and NS to show that
clear distinctions exist for these configurations in terms of radial variation
features. The LT precession in equatorial circular orbits increases with
approach to a BH, whereas for NS it increases, attains a peak and then
decreases. Interestingly, for $a_*=1.089$, it decreases until it vanishes at a
certain radius, and acquires negative values for $a_* > 1.089$ for a certain
range of $r$. For $1<a_*<1.089$, a peak appears, but the LT frequency remains
positive definite. There are important differences in accretion disk LT
frequencies for BH and NS and since LT frequencies are intimately related to
observed QPOs, these features might allow us to determine whether a given
rotating compact astrophysical object is BH or NS.
