Desktop applications often use a single database connection that persists. However, in PHP and most other web languages/frameworks this should not be the case. Instead of a single long lived connection, you use a short lived connection for each page load. Rather than thinking of your application as a whole, think of the process of each page load. In PHP, no other page load knows about the other unless you persist the data in a database or in $_SESSION or other places.
The singleton pattern in PHP is one of singleton throughout the page load. Although initially it may seem like "What's the point? Its just a page load", when you get to larger applications, it becomes obvious that each page load is not an insignificant operation. In systems like Drupal, you can have 10's to 100's of database queries happening each page load
Oh, and the singleton pattern is considered Bad Practice and will bite you later. In fact, it should probably be avoided in all of your applications. A better pattern would be something like the DAO pattern or the Factory pattern. Please, save yourself and don't use singletons (I've made that mistake so many times).
As for persisting objects between page loads (which is very necessary to almost any application), that is what your database and $_SESSION variable is for. You could also use alternative caching options like APC (http://php.net/manual/en/book.apc.php). Your application should be able to re-construct what it needs to based on whatever it finds in your database. For example, a user is created. You persist that user's data to the database. Whenever someone comes around and wants to know about the user, it pulls it out of the database to re-create your user object. Another example is that a user logs in. You want them to stay logged in, so you persist some kind of variable to $_SESSION that is checked every page load to see if they are logged in and who they are logged in as. You do some checking and then load the user that they say they are logged in as from your database.
