I want to point more than one domain name at my site. Absolutely, though it may cost quite a lot. If the name is on hold because it has not been paid for, it will generally become available 90 days after payment became overdue. If you find that the name is on hold for over 90 days, it is very likely the focus of a trademark or other dispute, and you will have to forget that name and try another. One hidden cost is the requirement to maintain a domain name server, or an arrangement with a company that provides a domain name server service.
Some companies offer a once-off fee, others charge every year.
Go for the former. The other hidden cost is not really hidden. However, it is easy to forget in the excitement of registering domain names, putting them on the market, promoting them etc. Although the domain name market is quite quiet now, it is bound to grow in the future. The supply of good domain names is limited.