Apple Mail keeps showing a spinning wheel

Not nearly enough information for us to help.

What Mac? What OS version?

What has recently changed?

Do other applications work, just not Mail?

Try restarting in Safe Mode. Does Mail work while in Safe Mode?

Please run Etrecheck and post its full report here. Use the "additional text" button and paste the report into the text box.

Part 1 of 2

Download the Application Etrecheck ( External Link ) directly from the Developer.

The Application is Not a " Silver Bullet " but only a tool to examine the Hardware / Software used on this computer 

This is a Diagnostic Tool that makes no changes to the computer.

It makes a coherent and readable inventory of both the Hardware and Software used on the computer 

The application is free or paid for added features. 

The Report will Not Reveal Any Personal Information. 

Post back the Full Report - copy and paste - >>>> using the Additional Text Icon  <<<<

Lastly - should you decide to use and post the Report now or at a later time, we are willing to have a very close look and offer some insights 

Part 2 of 2

At one time there was an  Etrecheck Application available on the Apple Apps Store.

Due to certain constrains place upon the Developer by APPLE, the developer chose to remove it from the Store and make is available Directly from them.

This specific application is well known,  on these Forums,  as a Go To for trouble shooting and avoiding all the question / answer and question / answer to finally get the issue pinned down.

It is also known and reported by Reliable Sources that,  sometimes APPLE SUPPORT has invoked using this same Developer Distributed Version of Etrecheck application to trouble shot computer issues 

Lastly - should you decide to use and post the Report now or at a later time, we are willing to have a very close look and offer some insights