In the numerical modeling of the propagation of elastic waves in unbounded domains, the physical domain is often truncated in order to reduce it to a finite computational domain, thus allowing the use of efficient domain discretization methods to simulate the wave motion. At the truncation boundary, Perfectly-Matched-Layers (PMLs) are used as wave-absorbing buffers, aimed at the rapid absorption/decay of the outgoing waves within the PML buffer, thus mimicking the far-field physical wave behavior, albeit at the physical domain’s truncation boundaries instead of at infinity. The wave-absorbing properties of the PMLs originate from their construction, which rests on a frequency-dependent coordinate transformation that turns the PML buffer into a Cosserat-like medium. The behavior of PMLs, including their long-time behavior, depends on the particular form of the stretching function used in the coordinate transformation. For example, it can be shown that the standard PML stretching function would always lead to instabilities in the time-domain that cannot be avoided, even though careful parameterization of the PML may delay their onset. The instabilities take the form of exponential growth that appear to originate suddenly from within the PML buffer, and, invariably, result in the pollution of the wave response within the physical domain. Using eigenvalue analysis (both frequency- and wavenumber-based), we demonstrate the origin of the instabilities, and their inevitability. Guided by that analysis, we discuss a stability condition that informs the PML parameterization, and when satisfied, could eliminate the instabilities. We further show that the instabilities associated with the standard PML can never be avoided, while for the Complex-Frequency-Shifted (CFS) PML satisfaction of the stability condition completely eliminates the instabilities.