If you don't see a featured class that you have posted to be visible on your public page, make sure that the class that you are trying to post as a featured item is in a time period that has been configured for online registration. Featured items will only appear on your public ASAP homepage if they are scheduled in a time period that is configured for online registration.
If you are not seeing any of your featured classes on your homepage, it could be because you have something in your Customize Your Homepage. Featured Classes and Customize Your Homepage may not be used at the same time. They occupy the same space, so choose to use one or the other. This includes any HTML text that might be remaining in the HTML tab in the Customize Your Homepage section. Make sure you have cleared the HTML tab in the Customize Your Homepage section of all text before you attempt to choose featured classes.
If you want to add copy to your homepage along with your featured classes, go to Configure/Online Registration Site and choose Add a Message to a Page/Homepage. There you can enter copy and it will appear along with your featured items/classes.