Both AWS and Azure will become quite expensive very quickly, and when you least expect it. That is to say, if it gets popular overnight you're going to be hurting in the morning.
If you're thinking about a public project that needs to be web facing and you're comfortable with Linux, DigitalOcean would be my choice. Drops you into a root shell with no mess or fuss. I've heard good things about Linode as well, but haven't used it personally. If you're not comfortable with that any commodity hosting will do, but your mileage will vary (they are all oversold and you will get fucked eventually).
If it's just something you need to stick up somewhere for only yourself, build out a virtual machine for it and host it locally.