As I noticed on your screenshot as well as on my sample file, the class column on the Customers & Jobs window would only show classes once the Job status is completed. You'll want to run and customize the Estimates by Job report to view the estimate transactions with associated classes. Here's how:
Go to the Reports menu and select Jobs, Time and Mileage, then choose Estimates by Job.
Click the Customize Report button.
In the Display window, search and select Class from the Columns section.
Hit OK. This will show you all estimate transactions with their associated classes by Job.
I'd like to share some insights about assigning classes on customer names. Normally, when your class preference is set to be assigned for name lists, the class has to be added on the customer profile. See the following screenshots:
If the class is not embedded to the customer profile, it will remain blank under the Class column. This is even when you assign the class to their respective sales transactions.
Don't worry, I have the steps to help you populate the Class info without opening the customer profile one by one. Here's how to do that:
Click the Customers menu then Customer Center.
Right-click on the Class header to reveal the options.
Select Add Edit Multiple Customer: Jobs.
Click the Customize Columns option at the top.
Scroll-down and click on Class then press Add. Tap OK after.
Find the customer with the blank Class information. Click the field for class in line with the customer to pick the correct one.
Use the Copy Down option if multiple customers will use the same classification. Do this by right-clicking the field with the class to copy.
When you're done, click the Save Changes button at the bottom.
I'm adding some Community links that highlights the processes in this post: