I often see people in online programming communities asking about cheap and easy ASP.NET Core hosting options. This post covers my thoughts on the matter, so that I don't have to keep writing the same answer over and over again! Generally, these posts are asking for cheap and easy options, so that is what I'll discuss.
Azure is your best bet - it's easy to use, and has the full weight of Microsoft behind it. Deploying can be as simple as a
git push. There is a free tier for 12 months. After that, you can get 'free forever' Azure App Service instances that have a few minor drawbacks (shared hosting, no custom domains, etc.), or you could pay for some of the cheaper shared instances. If you want more functionality, moving up to the other paid plans is definitely worth it, there are some great features there. Scott Hanselman has a great post - Penny Pinching in the Cloud - that covers the best way to get a good bang-for-your-buck from Azure App Service hosting.
The next best bet is to host on a DigitalOcean VM. You can use Dokku to get your own mini-Heroku PaaS, or manage the VM yourself (following Microsoft's documentation). You can get anywhere from $10 to $100 in credit from a referral link - this will last you a year and a half with a small VM.
Heroku is another good option. Their product is rock solid, easy to use, and has a wide variety of pricing tiers (including a free tier and an inexpensive hobby tier). You can deploy ASP.NET Core apps using a .NET Core buildpack, or using their Docker functionality.