what happens when simultaneous requests for the same database function is called? Is it possible that the database class will return the wrong information to other users accessing the same function at the same time?
Absolutely not.
Each PHP request is handled entirely in it's own process space. There is no threading, no application server connection pool, no shared memory, nothing funky like that. Nothing is shared unless you've gone out of your way to do so (like caching things in APC/memcached).
Every time the application starts, your Singleton will get created. When the request ends, so does the script. When the script exits, all of the variables, including your Singleton, go away with it.
What other similar problems could occur?
Unless you are using transactions (and if you're using MySQL, using a transaction-safe table type like InnoDB), it is possible that users could see partial updates. For example, let's say that you need to perform an update to three tables to update one set of data properly. After the first update has completed but before the other two have completed, it's possible for another process to come along and request data from the three tables and grab the now inconsistent data. This is one form of race condition.