Ideally, it would be nice if the database and chunks of it could be downloaded, not simply queried online. I have found several online databases but none that are downloadable. (Why is that?) One criteria for "best" would be a downloadable copy, but maybe that ideal cannot be realized at existing websites. What would you say is the best right now?
Here are highlights of what I am already aware of. CIRCL hash lookup is a big public database of SHA-256 hashes, but it's a specialized project. A free-for-all database would allow any string of length text (up to the max) to be referred to by their hash. For SHA-1 I would say sha1.gromweb.com is pretty good but is limited: Max string length is 255. As I write this they have 823,774,251 hashes in their database; however, they don't seem to offer a way to download it. hashes.com supports many hashes (e.g., MD5, SHA1, SHA256, SHA512), but they charge for queries and their database is proprietary. I like the website md5hashing.net. I like that they seem to rate-limit so both encodings and decodings take so much time it's impractical to use for cracking. They allow strings at least up to 406 characters. That's the length of the paragraph I submitted. The interface is simple. The text is stored by its MD5 hash. From that page many hashes are shown reversed. Again, though, one cannot download their database.