What is best os to run on iMac (21.5-inch, Mid 2010)? running MacOS Sierra 10.12.6 now

High Sierra is the latest version of macOS you can run.

How to upgrade to macOS High Sierra - Apple Support

Welcome!

Some data from you can help us give you a more targeted answer. Yours will run OS 10.13 but whether it runs well may depend on your specific hardware configuration. Example: some 2012 21,5-inch iMacs shipped with dual-core i3 processors that are not as robust as the optional dual-core i5 versions. We can also advise whether adding RAM is a good idea.

Please to "About this Mac..." from your Apple menu to get this:

On yours, please find and tell us what is listed for "Processor," "Memory," and "Graphics."

⚠️ For security reasons, please do not post your serial number.

You can view your memory capacity in the Memory tab

and /Applications/Utilities/Activity Monitor.app can show you how much memory you are actually using.

Ideally you want more physical RAM than you use, but if you need more when running multiple applications, the OS will dedicate space on the hard drive to swap out inactive memory for later use, that is the "Swap Space" and is much slower than actual physical RAM. As you can see, 12GB would not be enough in this case, and 16GB is the max for this machine that I use, and most of the time that is fine, but you have to be smart about not launching too many apps at the same time.

Also, if you are using your Mac to do your taxes, might be time to start shopping for a new one before next year's tax time...I just got this message after finishing mine on my 2010 iMac:

Pretty easy, use either Crucial.com or Macsales.com RAM. Both vendors will help you select the correct RAM. Installing it is a 5 minute job and only requires a screw driver. For example here is the correcthttps://eshop.macsales.com/installvideos/imac_mid_2010_mem/iMac11-2/

Your iMac has 4 RAM slots, 2 are currently empty. So buy 2 x 4 GB of RAM and install the new DIMMS in the empty slots.

and here are instructions on how to install it:

https://eshop.macsales.com/installvideos/imac_mid_2010_mem/iMac11-2/

You are welcome.

You can look up detailed specs including RAM configurations here:

https://everymac.com/systems/apple/imac/index-imac.html

OtherWorldComputing will have compatible RAM for your iMac.

You can occupy the two empty slots with a max of 4GB each in the 21.5" model. So, if you were keeping your existing 2x2GB RAM, you could operate with 12GB with 2,2,4,4 GB RAM chips, which should be reasonable performance,

https://eshop.macsales.com/item/OWC/1333DDR3S08S/

or replace the existing and max each slot out at 4,4,4,4, just depends on how much money you want to invest in the machine and how long you expect to remain using it. The RAM for that model is relatively older and somewhat cheaper compared to what the new machines use.

https://eshop.macsales.com/item/OWC/1333DDR3S16S/

Good luck with upgrading your machine.

Upgrade the RAM to 12 GB or more and then run High Sierra 10.13.6. Running with only 4GB of RAM is simply not enough.