First of all for those of you that are in a hurry, and need a fast solution go and look at this detailed starbound server hosting list with every provider possible and make your choice. Then come back here and leave a comment with your selection.
This is what you happens when you go with the wrong host. One day you get a message that the server has been down for the past 24 hours. You contact the support and you wait for a reply for at least 48 hours (this is a very good vilayer example, stay away).
A dedicated server usually the best option for any kind of server operations but sometimes like in the case of gaming, is really not worth it. Gaming server are not require that much power. Are well optimized and can run on a shared server very smooth. For example games like Counter Strike Go can run perfectly to a server that is shared with another 20 clients.
Be aware of overcrowded server
From the moment you select a shared server to host your games then you must be ready to face the downside of your choice. Some hosts do over sale the servers with bad results for your players experience. Servers will crash, random reboots, hackers will try to bring down your server, cheaters will ruin your game and your server might stay down for many hours if not days. The good side is that are always cheaper in comparison with a dedicated server. Also you no need to have any technical skills or be a coding geek. Good hosts are offering 24/7 support with no extra charges and phone support. If you pick the right host you can enjoy your game and stop worrying about server side issues.