Is it allowed to wear polarized sunglasses in cockpit?
This FAA document describes considerations for choosing sunglasses for pilots.

Since the document concludes, that


  Polarized sunglasses are not recommended


I presume there is no rule that would explicitly forbid it. However there is always the “operate aircraft careless and reckless manner” paragraph and if you badly screwed up while wearing such sunglasses and there was grounded belief that some of the disadvantages of such sunglasses contributed to the incident, you could be charged with that.

The document quotes following reasons why they are not recommended:


may reduce or eliminate visibility of instruments with polarization-based anti-glare filters,
may reduce or eliminate visibility of LCD instruments (LCDs emit polarized light by design),
may interfere with visibility through an aircraft windscreen by enhancing striations in laminated materials,
may mask the sparkle of light that reflects off shiny surfaces such as another aircraft’s wing or windscreen, which can reduce the time a pilot has to react in a “see-and-avoid” traffic situation.


Even if your plane does not have any instruments with polarization filters and no issues with striations in laminated windshield, filtering reflections from other aircraft is definitely not what you want.

Note that in most other situations, filtering reflections (from water, cars, metal roofs etc.) is desirable and is why polarized glasses exist, because reflections from close objects can be blinding. But in aircraft you are far from the sources of reflection, so the risk of blinding reflection is low, while the reflections can alert you to presence of other aircraft that might otherwise be hard to spot when looking towards the sun.