Arguably the most important shell extension is the text extraction (IFilter), which will help you find something among your PDF documents. And this is exactly the functionality you don't get for free from the most popular PDF readers (and some even get cheeky). Here is a summary of my findings, in the order of the program's alexa rank:
Table 1. Shell extensions offered by PDF programs
The Size column refers to the download size, but that's also indicative of the installed size which is even bigger. Look at the 3 most popular readers, which are also the heaviest; I don't know what they put in their installers, but it's definitely not shell integration. The absense of text filters isn't that important anymore since windows 8 and later include an IFilter component at the system level, to facilitate text search. Popular yes, sensible no. I recommend staying away from the likes of Adobe/Foxit/Nitro as all you get is lard and slow loading times.
Then comes PDF XChange Viewer that does everything, including text searches. That's why it is the recommended PDF plugin for xplorer². But I was pleasantly surprised with Sumatra PDF, which also does everything but at a fraction of the size. Yes, its previewer is rather basic (no zoom etc), but it is very lightweight and fast. So it is definitely worth a look. With software, size does matter but the inverse sense: less is usually more!
Another good thing with Sumatra PDF is that you can install just the shell integration options and continue using your existing standalone viewer, if you are hooked on adobe for some reason. When you install click on options button and tick everything except for the default PDF application checkbox.