Good question! I'll try an answer and hope it's a good balance between accuracy and what I guess is your (and other readers') knowledge of astrophysics.

[tl;dr: a neutron star (NS) and black hole (BH) in a close binary are practically guaranteed to "crash" into each other eventually. There seem to be lots of those binaries. Neutron star - black hole mergers (NSBH) from other possible arrangements (wider binaries, isolated individual NSes and BHs) will be vanishingly rare in "the local universe".]

There are two principal branches in an answer to your question.

First branch: there are probably lots of individual isolated neutron stars and black holes within galaxies, and those are unlikely to form binaries with each other, or get anywhere close to each other (except maybe possibly in the very centre of galaxies, or in the very far future). I won't spend time on this branch in this comment, even though this is the branch which interests me personally (at a theoretical rather than astrophysical level). In reality, isolated stars and star systems within galaxies just do not seem to collide with one another and isolated neutron stars and (stellar mass) black holes are no different.

Independent isolated stars may form binaries under rare circumstances such as galaxy-galaxy mergers or in radiating globular clusters within galaxies. The same goes for isolated neutron stars or stellar mass black holes. In those cases they can fall into the next paragraph.

Second branch: there are probably lots of neutron stars and black holes already in close orbit with one another, and those orbits will over long times decay (because such binaries radiate nontrivial gravitational waves) leading to a "crash". Tight orbits involving two relativistic bodies (white dwarf (WD), neutron stars, black holes) are unstable and practically always lead to a merger; there are a number of ways a looser/softer binary of such bodies can become a tighter/harder one, including during the transition of a dying star into a WD, NS, or BH.

Multi-star systems are commonplace[1] and likely mostly born as multi-star systems (e.g. within a nebula originating in the explosive demise of earlier more massive stars), rather than starting as individual stars that just get close enough to start orbiting each other.

One could compare this to how Jupiter and the sun most likely formed from the same part of a giant molecular cloud (GMC), rather than a fully-formed Jupiter being captured later by the sun. Likewise, the Alpha Centauri system probably formed as triple-star system within a different part of the same GMC, rather than the inner AB binary capturing Proxima Centauri (C) or each other. (Triple star systems with an outer member orbiting an inner binary are pretty common, even very close to us in the galaxy).

We'd expect therefore for a proportion of aging multi-star systems to evolve into binaries of "dead stars" including inner neutron star - black hole binaries. Those inner binaries will practically inevitably decay resulting in a neutron star - black hole merger (NSBH) producing a signal in-principle detectable by gravitational wave detectors like KAGRA, Virgo, and LIGO. (The outer member of a triple or surrounding gas, dust, asteroids, and comets can help tighten the inner binary.)

Gravitational wave detectors like LIGO are range-limited, with the current range about 200 megaparsecs (about 600 million light years), which is on cosmological scales quite local. We've counted about a quarter of a million galaxies[2] in that range. By comparison there are hundreds of billions of galaxies in our sky.

A fraction of neutron stars are pulsars. Because of the practicalities of radiotelescopes, at present we can only really find pulsars nearby within our galaxy (dust obscures a lot of the galaxy at the relevant radio frequencies, and there is only one extragalactic pulsar known, and it's in the neighbouring Large Magellanic Cloud). However, we've found several thousand pulsars (see final paragraph below). A sizable fraction are in multi-star systems, which we know from tracking radial acceleration in several ways, and because among these are a handful of eclipsing binaries. There are known pulsar-pulsar pairs (famously, <https://en.wikipedia.org/wiki/Hulse%E2%80%93Taylor_pulsar>) as well as more complicated systems which include white dwarfs and maybe black holes.

From gravitational wave observations, we can spot a NSBH even if we can't see either individual body (in radio or other parts of the electromagnetic spectrum) because of obscuration by dust and other foreground matter, or because the NS is not a pulsar and thus both objects are very faint in radio. We don't have a good census of dim neutron stars even in our part of the galaxy. However, there will be more dim neutron stars than radio-bright pulsars, and it is reasonable to guess that there will be more dim neutron stars in binary- and multi star systems with a black hole than there are radio-bright pulsars in such systems.

So looking at one galaxy (ours), we can guess that if we waited a few million years we would see a decaying hard neutron star/black hole binary eventually merge. So naively, we might think that if we waited a few years we'd see an NSBH merger from among about a quarter of a million galaxies. And at gravitational wave detectors we're seeing more NSBHs than that naive estimate. So perhaps pulsarNS-BH binaries are rare compared to nonpulsarNS-BH ones. Or perhaps there are lots of pulsarNS-BH binaries in the Milky Way hidden behind dust and sources of radio noise.

Finally, this interactive map of known pulsars https://ishivvers.github.io/maps/pulsars.html is so cool I've submitted it separately at https://news.ycombinator.com/item?id=39970373

[1] "Stellar Multiplicity" (arxiv version of published ARA&A paper), Duchene et al. 2013 https://arxiv.org/abs/1303.3028

"Multiple Star Systems" (multimedia outreach) https://science.nasa.gov/universe/stars/multiple-star-system...

(examples) https://en.wikipedia.org/wiki/Category:Triple_star_systems

(more examples, quadruples and beyond)
https://en.wikipedia.org/wiki/Star_system#Quadruple

[2] section 5.3 of "Census of the Local Universe [..]", Cook et al. 2019 <https://ui.adsabs.harvard.edu/abs/2019ApJ...880....7C/abstra...>