If a customer is on a waitlist for a class and a space in that class opens up, the waitlisted customer is not informed of an opening automatically, ASAP leaves that to the Admin to contact the person and find out if they still want to enroll.
As long as you don't allow your students to transfer classes on your public ASAP website, the only people conducting drops and transfers will be users on your Admin site. This gives you the most control over waitlists because Admin users will know when students are dropped/transferred out of a class and when space opens.
Best practice is to note the number of students on a waitlist when you are dropping or transferring students out of a class. After you drop or transfer the student, select the Waitlisted tab in the Class Details screen and notify the waitlisted student that space is open. If you are concerned that the space in the class might be taken by a new registrant, you can enroll the waitlisted student to secure the space for him/her, making sure that you collect payment from the student or drop the waitlisted student and inform/register the next waitlisted student.
From the Class Details page, select the waitlisted student...
This will take you to the Student - Class Detail page. From here, select Register from Waitlist to enroll the student in the class and remove him/her from the waitlist:
Students who are enrolled in a class but haven't paid can log into their accounts on your Public site and make payment there.
Another way for an admin to review openings that have customers waiting, is to go to View Reports|Class Summary, and run the report, which will list all classes, capacity, enrollments, watilisted, and open slots. If you find classes with open slots and waitlists, contact students on the waitlist and let them know about openings.