Supabase, Firebase, and other cloud-based databases should work when your project has the correct connection information and is configured correctly.Local databases can work, but saving changes over time may not work as expected and should be used with caution.
For simple websites, such as blogs or other content sites, create a static site. You can host these entirely on Netlify.
For web apps, follow Netlify’s guidance on Composable Architecture. Build a frontend that you host on Netlify, with a decoupled backend using Supabase for your database, and a headless CMS if you need content-editing abilities. Supabase provides an API to interact with your database, as well as authentication and storage services.