I noticed that most sites send the passwords as plain text over HTTPS to the server. Is there any advantage if instead of that I sent the hash of the password to the server? Would it be more secure?
No, in fact this would be a vulnerability. If the attacker is able to obtain the hash from the database, then he could use it to authenticate without needing to crack it. Under no circumstance should a user or an attacker be able to obtain a hashes password.
The whole point of hashing passwords is to add an extra layer of secuirty. If an attacker is able to obtain the hash and salt from the database using SQL Injection or an insecure backup then he has to find the plain text by brute forcing it. John The Ripper is commonly used to break salted password hashes.
Not using https is a violation of the OWASP Top 10: A9-Insufficient Transport Layer Protection
EDIT:
If in your implantation you calculate a sha256(client_salt+plain_text_password)
and then calculate another hash on the server side sha256(server_salt+client_hash)
then this is not a serious vulnerability however it is still susceptible to ease dropping and replaying the request. Thus this is still a clear violation of WASP A9. However, this is still utilizing a message digest as a security layer.
The closest thing i have seen to a client-side replacement for https is a diffie-hellman in key exchange in javascript. However, this does prevent active MITM attacks and thus is till technicality a violation of OWASP A9. The Authors of the code agree that this is not a complete replacement for HTTPS, however it is better than nothing and better than a client-side hashing system.
Since it's over HTTPS, it's definitely just fine to send the password without hashing (over HTTPS it's not plaintext). Furthermore, if your application is depending on HTTPS to keep it's content secure, then it's useless to hash the password before sending it over HTTPS (i.e. if an attacker can unencrypt the data on the wire, you're screwed anyways)
It would actually be less secure to hash the password and send it over a non-encrypted channel. You will expose your hashing algorithm on the client. Hackers could just sniff the hash of the password and then use it to hack in later.
By using HTTPS, you prevent a hacker from obtaining the password from a single source, since HTTPS uses two channels, both encrypted.
If you're connected to an https server the data stream between the server and browser should be encrypted. The data is only plain text before being sent and after being recieved. Wikipedia article
Use HTTP Digest - it secures the password even over http (but best useage would be http digest over https)
Wikipedia:
HTTP digest access authentication is one of the agreed methods a web server can use to negotiate credentials with a web user (using the HTTP protocol). Digest authentication is intended to supersede unencrypted use of the Basic access authentication, allowing user identity to be established securely without having to send a password in plaintext over the network. Digest authentication is basically an application of MD5 cryptographic hashing with usage of nonce values to prevent cryptanalysis.
Link: http://en.wikipedia.org/wiki/Digest_access_authentication
If you want to see a "real life" use, you could look at phpMyID - a php openid provider that uses http digest authentication http://siege.org/phpmyid.php
.. or you could start from the php auth samples at http://php.net/manual/en/features.http-auth.php
Http digest rfc: http://www.faqs.org/rfcs/rfc2617
From my tests all modern browsers support it...
Sending a hash over the wire completely defeats the purpose of the hash, because an attacker can simply send the hash and forget about the password. In a nutshell, a system that athenticates using a hash in clear text is wide open and can be compromise with nothing more than network sniffing.
If you're looking to replace a clear-text password over HTTPS with a hashed password over HTTP then you're asking for trouble. HTTPS generates a random, shared transaction key when opening up a communication channel. That's hard to crack, as you're pretty much limited to brute forcing the shared key used for a (relatively) short-term transaction. Whereas your hash can be just sniffed, taken off-line and looked up in a rainbow table or just brute forced over a long amount of time.
However, a basic client-side password obfuscation (not hash) sent over HTTPS does have some value. If I'm not mistaken this technique is actually used by some banks. The purpose of this technique is not to protect the password from sniffing over the wire. Rather, it's to stop the password from being usable to dumb spying tools and browser plug-ins that just grab every HTTPS GET/POST request that they see. I've seen a log file captured from a malicious website that was 400MB of random GET/POST transactions captured from user sessions. You can imagine that websites that used just HTTPS would show up with clear-text passwords in the log, but websites with very basic obfuscation (ROT13) as well would show up with passwords that are not immediately of use.