  Gravitational waves can probe the existence of planetary-mass primordial
black holes. Considering a mass range of $[10^{-7}-10^{-2}]M_\odot$,
inspiraling primordial black holes could emit either continuous gravitational
waves, quasi-monochromatic signals that last for many years, or transient
continuous waves, signals whose frequency evolution follows a power law and
last for $\mathcal{O}$(hours-months). We show that primordial black hole
binaries in our galaxy may produce detectable gravitational waves for different
mass functions and formation mechanisms. In order to detect these inspirals, we
adapt methods originally designed to search for gravitational waves from
asymmetrically rotating neutron stars. The first method, the Frequency-Hough,
exploits the continuous, quasi-monochromatic nature of inspiraling black holes
that are sufficiently light and far apart such that their orbital frequencies
can be approximated as linear with a small spin-up. The second method, the
Generalized Frequency-Hough, drops the assumption of linearity and allows the
signal frequency to follow a power-law evolution. We explore the parameter
space to which each method is sensitive, derive a theoretical sensitivity
estimate, determine optimal search parameters and calculate the computational
cost of all-sky and directed searches. We forecast limits on the abundance of
primordial black holes within our galaxy, showing that we can constrain the
fraction of dark matter that primordial black holes compose, $f_{\rm PBH}$, to
be $f_{\rm PBH}\lesssim 1$ for chirp masses between $[4\times
10^{-5}-10^{-3}]M_\odot$ for current detectors. For the Einstein Telescope, we
expect the constraints to improve to $f_{\rm PBH}\lesssim 10^{-2}$ for chirp
masses between [$10^{-4}-10^{-3}]M_\odot$.
