The standard way I've seen is to do this when the user logs in: if in checking their password you notice they're using the old hashing scheme, you use the plaintext password provided by the login process to generate a hash using the new scheme.
For example, lets say you're a PHP function that takes in $username and $password (both plaintext). You're "upgrading" from unsalted MD5 to unsalted SHA1 (NOTE: DO NOT DO THIS IN REAL LIFE. READ http://codahale.com/how-to-safely-store-a-password/ AND UNDERSTAND WHY UNSALTED MD5 / SHA1 SHOULD NOT BE USED FOR PASSWORD SECURITY). Your code to upgrade the hashes would look something like this:
if (md5($password) === $cur_user['password'])
$cur_user['password'] = sha1($password);
In this case, the problem is that passwords longer than eight characters are silently truncated. So, if my password is something like supersecretpassword, I could also enter supersec and have it accepted (since they hash to the same value). That would be a problem for Amazon, since someone could make a typo in their password and have the wrong value stored in the new hash.
Not really. It's more of a fundamental property of hashing functions, and it's technically true of any hashing function, and collisions are nearly just a technical possibility unless you're using an incredibly-small hash output. Though that would likely be preferable to truncating before hashing - fewer practical collisions and safer, as reversing the hash is less useful due to more collisions actually existing in breakably-short password lengths.
1. They could force everybody with an old password to reset it. This would announce there was a flaw, but shouldn't be any riskier than having a "forgot password" link.
2. Silently upgrade passwords as users log in. This is risky, because many users might not log in for months.
3. Use their clusters to reverse weak password digests. This would be mostly transparent to customers, but runs a risk of accidentally locking out some customers. For example, if "bar" and "foo" both crypt() to "abcd123", and the user has password "foo", the recover-and-redigest method might change the password to "bar".
4. Wrap old passwords in a new digest -- presumably they have some way to determine which algo is used per-user. They could treat the output of crypt() as the input to a more secure digest, like scrypt, and then just store "crypt-plus-scrypt" in their database as the digest type.