Is the output of a laser pointer polarized or not? How to verify it? 
Is a laser necessarily polarized?
I have a laser pointer with wavelength 532nm in hand.
 A: There are two main types of diode lasers that dominate the commercial market. Edge-emitting diode lasers have very asymmetric cavities, which leads to polarization with electric field in the "vertical" direction (normal to the surface of the chip), IIRC. The other type, VCSELs, have a much more symmetric cavity and could be polarized in either direction (although they usually have enough asymmetry to select two preferred polarization axes). 
The most common way to produce a 532-nm laser beam is to use a 1064-nm laser diode with a frequency doubling element. 
1064-nm lasers are available as both edge-emitters and VCSELs. I expect (but don't know for sure) that a laser pointer is more likely to use an edge-emitter. If you have the spec for the diode itself, it's easy to tell because for edge-emitters the beam divergence is much wider in the vertical direction. In a laser pointer it is probably more difficult to tell which direction is which because even if the original source is an edge-emitter they will use optics to circularize the beam.
A: No, laser light is not necessarily polarized.
There are many different types of lasers.  In gas lasers, many different modes may be excited in the cavity, but only modes that are not very lossy end up being amplified and emitted - this can result in a single polarization.  However, there may be multiple modes with different polarizations that have this property, resulting in a more random polarization.  There also may or may not be a polarizing filter built in to the laser aperture.  I don't know as much about other types of lasers, but I don't think they all are necessarily polarized.
You could test whether or not your laser is polarized by getting a polarizing film (for example polarized sunglasses).  Shine the laser through the film while rotating the film.  If the intensity of the transmitted beam changes when you rotate the film, then it is at least partially polarized.
A: The reason I am adding this answer is that I googled "polarized laser" and was directed here, because the laser I recently bought is also 532nm and I discovered more or less by accident that this Chinese-manufactured laser emits plane-polarized light.
The effect is pretty pronounced. I didn't bother to quantize the result (maybe I will edit this later) but for proper orientation of a polarizing film almost all the light is passed/blocked for reasons that I think The Photon has explained.  
A: Laser light is polarized most of the time,while it is unpolarized some of the time.I think you should look at this to make it more clear.
