Good domain name is a name that either clearly describes what the website is about or is some easy to remember word phrase. In both cases finding a free domain is far from easy.
You can easily find that most popular words or word phrases under the com TLD are already taken. Trying org or net top level domains is one option but most like to stick with the well known com name space.
Before you start with various options to find a free domain you should check if there is some suitable domain for sale. There are many web places that offer registered domains for few 10 dollars up to ridiculously high amounts. In fact, many registered domains have no website. They were registered with only one purpose: to be sold for a lot of money. Of course, this “a lot of money” rarely comes because there are many other options how to get a good domain paying only the registration fee.
If you are looking for a domain that describes your niche, product or service and all general purpose top level domains are already taken then you can try to slightly modify your word phrase. For example, you can add words like “shop”, “site” or “info” to make a slightly longer domain name that is hopefully free. Another approach is to add a word before the target phrase. Choose a word that will emphasize the content of the website and will make a domain that is not registered yet.
If you don’t have some particular word phrase in mind and you only need some short, easy to remember domain name then you need a different approach. You will still find that almost all possible word combinations are already registered. One of the easiest ways to find a suitable and available word combination is to use domain suggestions tool like the one at Domain Tools.
You start with arbitrary word and then the tool creates various word combinations and checks domain availability. After some experimentation you will find some word phrases that will sound good and will be free for registration. In the case you would like to check a longer list of possible domains you can use the bulk check function where each suggested domain will be checked for availability with few popular top level domains.
Finding a good domain takes time. It is pretty easy to find some suitable domain that is registered but available for sale. But if you are not willing to spend more money than the registration fee then you will have to go through various options until you find a domain that is free.