Disclaimer: Don't take my answer as an absolute truth. It reflects only my current knowledge.
OK, I know roughly how the distribution of stars breaks down in terms of spectral type (lots of M stars, very few O/B/A stars, and the rest in between).
But is there any data that tells us the distribution of star sizes (ie I, II, III, IV, V, VI, D) in the milky way?
(If you already now about star populations, skip this)
The Milky Way is not composed only of the spiral arms and the core. We can divide the Milky Way rougly in three parts: the disk, the halo and the buldge.
The disk is comprised of the spiral arms and almost everything that lies in the plane of the galaxy. The sun (Sol) belongs to the disk.
The halo is more or less a sperical ball of very old stars (usually grouped together in Globular Clusters), that has a radius around 50,000 ly or more, engulfing the disk. The main features of the stars that belong to the halo is: Either they cross the disk at very large transversal speeds or they lie very far from the disk. They are very old, since the Globular Clusters aren't massive enough to hold enough gas to continue star formation.
The buldge is the core of the Milky Way, with a lot of gas and a very huge black hole in the center.
Astronomers usually call stars in the disk and buldge as Population I stars, while Population II stars are stars that lie in the Halo.
(
It would be nice to post a picture here with the structure of our Milky Way)
Answering your question, in the Halo (population II stars) you will only find very old stars, so no O/B/A/F main senquence stars and possibly no supergiant stars (except perhaps of some scattered stars from the disk), since the most massive stars from Globular Clusters leaving the main sequence are probably not massive enough to become supergiants. Globular Clusters are full of white dwarfs, but they are very difficult to be detected, due to their little luminosity and the great distances. On the other hand, you will find a lot of newly born main sequence stars in the disk and relatively much more supergiants compared to the Halo, since in the disk still exist gases to create new stars (possibly the gas that the Globular Cluster lost went to the disk and to the buldge).
Looking at the RECONS nearest stars list, it seems that there are about 135 known stars within about 7 pc of Sol, and about 125 (92%) of them are V, 8 (6%) are white dwarfs, and only 2 stars (less than 2%) are IV. Is this a typical distribution?
Honestly I don't know. But it might be biased, because white dwarfs are very dim, and difficult to be seen. Even though hot white dwarfs (with effective temperatures above 15,000 K are probably easy to be spotted by RECONS, the same doesn't hold true for very cold white dwarfs (recently they have discovered some white dwarfs with very low temperatures (around 3,000 K)).
It's hard to get a precise distribution of stars, even in our neighbourhood, because low mass stars and brown dwarfs are very hard to spot. Incidentally, Proxima Centauri is the nearest star to the Sun, but being a M5V star about 10,000 times less luminous than the Sun it was discovered only in 1915.
It's even possible that brown dwarfs outnumber main sequence stars, and the number of dead stars (white dwafs and less numerous neutron stars and black holes) is rougly equivalent to the number of main sequence stars in our stellar neighbourhood, but they remain undetected due to their intrinsic very low luminosity.
I had a link to a page with the distribution of main sequence stars in our neighbourhood. If if find it, I can post here.
I'm not a professional astronomer, and I would also like to now this answer. If you (or someone else) find something related or more precise information, please post here.