Circular polarizers are the ONE filter that's still very useful, in the age of digital imaging. C-Pol can do things that are impossible to replicate in post processing (... and NO, you cannot remove reflections with software... it's much more than just water and glass, too... there can be reflections off foliage in scenic shots or even off skin in portraits).
Yes, a C-Pol helps with contrast and color. It also reduces the effects of haze.
I carry a set of good C-Pol to fit all my lenses... But I mostly use them with wide angle, normal and short telephotos. Rarely with longer teles. I don't use one all the time, but do use C-Pol far more often than any other type of filter. Here's an example where I used a C-Pol to enhance the blue of the sky, make the white of the fishing boats "pop" and reduce (but not completely remove) the reflections off the water...
It might be easier to tell or show you when NOT to use one...
I won't use a C-Pol when the light is low and I can't afford the loss of light. A standard C-Pol reduces light between approx. 1.5 and 2.25 stops, depending upon how strongly its effect is dialed in. HOWEVER, today there are "High Transmissive" C-Pol available that reduce light less... approx. 3/4 stop to 1.5 stops. Those can help in some cases. I have one and haven't noticed any less polarization effect with the high transmissive type, compared to standard C-Pol.
Another time that I won't use a C-Pol is when shooting a sunrise or sunset directly. In fact, I remove any and all filters in those instances. But a C-Pol in particular, because they are multi-layered and more likely than most to cause added flare issues in images when there's a strong light source (like the setting or rising sun) within or nearly in an image. Also, a C-Pol serves absolutely no purpose when pointed directly at the sun. It has no effect.
Below is a test shot I took where I deliberately used a filter to see how bad the flare issues would be.
In the above, even using a high quality, multi-coated filter, there's overall "veiling flare", as well as some flare artifacts or "ghosts". Some aren't visible at Internet sizes and resolutions. There also was chromatic aberration that may have been accentuated by the filter. In this case I was able to make adjustments and retouch away most of the problems in post-processing (as shown below), but that was extra work that was unnecessary and could have been avoided simply by removing the filter!
Here's another example where I removed any and all filters...
Also, there are times when the reflection is an important aspect of an image, so the last thing I'd want to do is reduce that (as in the colorful reflections of the sunset off the wet sand and rocks, in the above image).
Finally, C-Pol can be tricky with wide angle lenses. The filter is most effective at 90 degrees from the light source. This can cause uneven effects, especially when a lens makes for a wide angle of view. Here's an example where the sun is off to the left, so the filter's effect is stronger toward the right hand side of the image...
The uneven effect of the filter can be seen in the viewfinder before taking the shot... and can actually be useful. In the image above, the cloudless blue sky was a bit boring, so I sort of liked the uneven polarization and decided to use it. I framed the image to use the deepening effect of the filter around the primary subject.
Below is another image where a C-Pol's effect was uneven (stronger on the left this time), but is much less noticeable thanks to the clouds and other detail in the image...
I recommend spending enough to get a very high quality, multi-coated filter (or filters, plural, if different sizes are needed... I don't like to use step rings because with most lenses they make it impossible to use a well-fitted hood, which I also consider very important, perhaps even more-so when using a filter).
I mostly use B+W F-Pro MRC (8-layer multi-coated) or XS-Pro MRC (16-layer multi-coated), both of which are now High Transmissive and which are a really good value. Prices vary by size, but those B+W are among the best, top-to-of-the-line filters. They use German Schott glass, brass frames, etc.... and yet they commonly sell for about half the price of other, similar specification filters (Breakthrough X4, Heliopan SH-PMC, Hoya HD3). They're even less expensive than a lot of filters that are lower specification. Hard to beat!
Circular polarizers are the ONE filter that's stil... (