How common are neutron stars? I thought this would be a rather easy question to answer, but it doesn't seem to be addressed anywhere. I haven't even been able to find a paper, article, or page that says it's unknown how common they are. I understand neutron stars, not in the form of pulsars/magnetars would be difficult to detect, but their precursors, 10-29 solar mass stars should be rather easy. Then, couldn't we infer from there? My ultimate question is the ratio of black holes to neutron stars as a result of core collapse. 
 A: TL;DR A crude estimate would be that as many as 0.26% of stars in the Galaxy are neutron stars and 0.06% are black holes, but these figures are highly uncertain.
Details
Neutron stars are almost undetectable once they have gone through the short-lived (10 million years or so) pulsar phase, so just counting neutron stars isn't going to give an answer.
You also can't just count the number of those stars you assume to be progenitors because they were massive, short-lived and hence sample only the last $\sim 20$ million years of the Galactic star forming history. The star formation rate would have been different (higher) in the past and our census of massive stars is highly incomplete due to dust in the Galactic plane where they form.
Let us instead assume that $N$ stars have ever been born in the Milky Way galaxy, and given them masses between 0.1 and 100$M_{\odot}$. Next, assume that stars have been born with a mass distribution that approximates to the Salpeter mass function - $n(m) \propto m^{-2.3}$. Then assume that all stars with mass $m>25M_{\odot}$ end their lives as black holes and all stars with $8<m/M_{\odot}<25$ end their lives as neutron stars.
So, if $n(m) = Am^{-2.3}$, then
$$N = \int^{100}_{0.1} A m^{-2.3}\ dm$$
and thus $A=0.065N$.
The number of black holes created will be
$$N_{BH} = \int^{100}_{25} Am^{-2.3}\ dm = 6.4\times10^{-4} N$$
i.e 0.064% of stars in the Galaxy become black holes. NB: The finite lifetime of the Galaxy is irrelevant here because it is much longer than the lifetime of either black hole or neutron star progenitors.
In a similar way, the number of neutron stars
$$N_{NS} = \int^{25}_{8} Am^{-2.3}\ dm = 2.6\times10^{-3}N$$
Now we use these results as scaling factors to apply to any stellar population. For instance, there are about 1000 "normal" stars in a sphere of 15 pc radius around the Sun, thus a density of 0.07 pc$^{-3}$. One can use the results above (using 1000 as the total number of stars born and ignoring the $\sim 10$% that have died) to calculate the density of compact remnants and then take $(3/4\pi n)^{1/3}$ as an estimate of the average distance to one of them. This gives an expectation value of 18 pc to the nearest black hole and 11 pc to the nearest neutron star.
The distance calcuated to the nearest black hole and neutron star remnants is likely to be an underestimate because many escape from the Galaxy or have very high velocity dispersions and much larger Galactic scale heights than normal stars. In other words, my density fractions quoted above are likely overestimates if applied to a local or any disk population in the Galaxy.
They are probably better estimates when looking at the Galaxy as a whole to calculate the total populations of black holes and neutron stars, but then there is considerable (factors of a few) uncertainty on the correct value for $N$.
Finally, it should be pointed out that there are uncertainties both on the index for the mass function I used (and whether it is invariant with time and Galactic location) and in the physics that determines the upper and lower limit masses to the integrals. In general, the upper limit uncertainty does not affect the estimates too much, but the lower limit is important. For neutron stars, the $8M_{\odot}$ figure I've used could be anywhere from $7-10M_{\odot}$.
The $25 M_{\odot}$ upper limit for neutron stars is however also the lower limit for black holes and this is highly uncertain. That means any estimate for the ratio of black holes to neutron stars (the above estimates say that ratio is about 1/4) is also highly uncertain and hugely sensitive to the assumed boundary between the initial masses of their progenitors. This in turn probably also depends on metallicity and rotation rate.
A: I guess that using the initial mass function (IMF) would yield a rough estimate of the number of neutron stars in a region.
The initial mass function $\xi (m)$ describes the inital mass distribution of a population of stars, so if you make an assumption on the mass of stars that will evolve into neutron stars, you would be able to determine an estimate of the number that you are looking for.
For example, you may assume that stars which initial mass lies between $M_1 = 8~M_{\odot}$ and $M_2 = 40~M_{\odot}$ are very likely to evolve into neutron stars as a result of core collapse. Then, the number $N$ of neutron stars that you would find in a region is given by:
$N = \int^{M_2}_{M_1} \xi (m) dm$
Where $\xi (m)$ is the initial mass function that has been measured in that particular region (please note that studies tend to show that the IMF may be universal).
However, the result will obviously depend on the choice of $M_1$ and $M_2$, and thus on our knowledge of the mechanisms of formation of neutron stars. Moreover, I doubt that this method is really precise, as it does not take into account every scenario that could lead to the formation or destruction of a neutron star (like the coalescence of two neutron stars that would end up in the creation of a black hole, or the formation of a new neutron star by accretion on a white dwarf in a binary system)
Just my thoughts here, please feel free to comment.
