Why aren't there spherical galaxies?

This whole question is a mistaken premise. There are spherical (or at least nearly spherical) galaxies! They fall into two basic categories - those elliptical galaxies that are pseudo-spherical in shape and the much smaller, so-called "dwarf spheroidal galaxies" that are found associated with our own Galaxy and other large galaxies in the "Local Group".

Of course when you look at a galaxy on the sky it is just a two dimensional projection of the true distribution, but one can still deduce (approximate) sphericity from the surface brightness distribution and large line of sight velocity distribution for many ellipticals and dwarf spheroidals.

Dwarf spheroidal galaxies may actually be the most common type of galaxy in the universe.

These galaxies are roughly spherical because the stars move in orbits with quite random orientations, many on almost radial (highly eccentric) orbits with no strongly preferred axes. The velocity dispersion is usually much bigger than any rotation signature.

There is an excellent answer to a related question at Why the galaxies form 2D planes (or spiral-like) instead of 3D balls (or spherical-like)?

Pretty pictures: UK Schmidt picture of the Sculptor dwarf spheroidal galaxy (credit: David Malin, AAO)

The E0 elliptical galaxy M89 (credit Sloan Digitized Sky Survey).

Details: I have found a couple of papers that put some more flesh onto the argument that many elliptical galaxies are close-to spherical. These papers are by Rodriquez & Padilla (2013) and Weijmans et al. (2014). Both of these papers look at the distribution of apparent ellipticities of galaxies in the "Galaxy Zoo" and the Sloan Digitized Sky Surveys respectively. Then, with a statistical model and with various assumptions (including that galaxies are randomly oriented), they invert this distribution to obtain the distribution of true ellipticity $\epsilon = 1- B/A$ and an oblate/prolate parameter $\gamma = C/A$, where the three axes of the ellipsoid are $A\geq B \geq C$. i.e. It is impossible to say whether a circular looking individual galaxy seen in projection is spherical, but you can say something about the distribution of 3D shapes if you have a large sample.

Rodriguez & Padilla conclude that the mean value of $\epsilon$ is 0.12 with a dispersion of about 0.1 (see picture below), whilst $\gamma$ has a mean of 0.58 with a broader (Gaussian) dispersion of 0.16, covering the whole range from zero to 1. Given that $C/A$ must be less than $B/A$ by definition, this means many ellipticals must be very close to spherical (you cannot say anything is exactly spherical), though the "average elliptical" galaxy is of course not.

This picture shows the observed distribution of 2D ellipticities for a large sample of spiral and elliptical galaxies. The lines are what you would predict to observe from the 3D shape distributions found in the paper.

This picture from Rodriguez and Padilla show the deduced true distributions of $\epsilon$ and $\gamma$. The solid red line represents ellipticals. Means of the distributions are shown with vertical lines. Note how the dotted line for spirals has a much smaller $\gamma$ value - because they are flattened.

Weijmans et al. (2014) perform similar analyses, but they split their elliptical sample into those that have evidence for significant systematic rotation and those that don't. As you might expect, the rotating ones look more flattened and "oblate". The slow-rotating ones can also be modelled as oblate galaxies, though are more likely to be "tri-axial". The slow rotators have an average $\epsilon$ of about 0.15 and average $\gamma$ of about 0.6 (in good agreement with Rodriguez & Padilla), but the samples are much smaller.

Actually, there are parts of a galaxy that extend beyond the galactic plane:

• Galactic halo: This is actually the primary part of a galaxy that is not in the main galactic disk. It's made up of multiple sections, and is composed or an array of objects.

  • Dark matter halo: This is a section of the galaxy's dark matter that exists in a semi-spherical shape. We can figure out the size and shape of the halo (though it's typically spherical) through its effects on the large-scale motion of stars.

  • Galactic spheroid: This is a region near the center of the galaxy made up of stars with odd orbits. I think of them as sort of like the comets in the Kuiper Belt - following odd, 3D orbits. The stars might have been perturbed by the central black hole in the galaxy - in our case, Sagittarius A*.

  • Galactic corona: Bits of gas and dust that follow irregular paths across the galaxy. They interact with matter inside the galactic disk and thereby oscillate around.

• Galactic bulge: This is the central part of the galaxy, around the central supermassive black hole. They're composed of gas, stars and dust.
• Stellar stream: A series of stars that have interacted with another object gravitationally. They may be the remnants of a dwarf galaxy.

I list these as examples to show that not all objects stay in the galactic plane. The other answers should give you an idea of why most objects do stay in the plane.

All matter in the galaxy has to rotate (not necessarily in the same direction) so that a centrifugal force acts. Without the centrifugal force, all matter contained in the galaxy will collapse into the center of the galaxy due to gravitation. The rotation happens about an axis, a line about which all matter revolves in the galaxy. Now, the manner in which all the matter revolves around that axis is planar. Why is it planar and why does it have to rotate about an axis only? The answer to this question will decisively clear that doubt.

But how does the planar galaxy continue to retain planarity for billions of years?

Let's imagine that a planar galaxy has a few bodies which don't revolve around the central axis and have their own axis of rotation. In any direction perpendicular to that axis, centrifugal force keeps the body from collapsing into the center of the galaxy. In any direction parallel to that axis however, there is no such centrifugal force; but there is a component of the gravitational force from the matter contained in the planar galaxy below. This component of gravitational force keeps pulling the body toward the plane, and there is no force to stop it. Thus, even this body will eventually join the galactic plane. All such fringe bodies which do not obey to the galactic plane will be attracted by gravity to eventually join the plane. Therefore the galaxy manages to maintain planarity.

As Rob Jeffries pointed out, there are galaxies that are of spherical and other three-dimensional shapes. There, however, since there is no pre-existing plane of rotation, nothing is causing the matter to collapse into a plane. Therefore, those galaxies retain their three-dimensional shape.