Well, $con will already be an object, as it's instantiating a new PDO object. Unless you're trying to add functionality to your PDO object, wrapping it is pointless.
That said, the best way to share your userdb/PDO object (depending on whether or not you stick with the wrapper) with other objects is to use Dependency Injection. That's a fancy term for passing your db to whatever object needs it. Since objects are defaultly passed by reference in PHP, if you create your db object first, all objects that receive it as a constructor/method argument will be using that same single instance.
EDIT: Link to Dependency Injection implementation
EDIT2: Clarification on DI in small projects -
The normal DI pattern generally requires a special object called a DI Container. This is a special use object that automatically injects the dependency into the object that needs it. For small projects, it's overkill. The simple, low complexity version of DI is simply:
class SomeClass {
protected $db;
public function __construct($db) {
$this->db = $db;
}
}
class SomeClass2 {
public function SomeMethod($db) {
// do something with the db
}
}
$db = new PDO(/* connection string */);
$obj = new SomeClass($db, /* other constructor args */);
// or
$obj2 = new SomeClass2(/* constructor args */);
$obj2->someMethod($db, /* method args */);
The magic is that since objects are passed by reference by default in PHP, $obj and $obj2 are using the same db connection.
The whole idea is to not blow apart scope or encapsulation by having a static method, and to ensure that classes and their methods are up front about what they require in order to work.
Singletons do the exact opposite. They're accessed through static methods, which bypass scope, and since they're invoked and not passed, they never show up in method signatures, so anyone not familiar with the code won't be aware of that hidden requirement. Even Erich Gamma, one of the people who helped codify the Singleton Pattern has regrets about it:
I'm in favor of dropping Singleton. Its use is almost always a design smell.
In PHP, where there's no concept of shared memory, and where scripts execute once per request, the only reason to want to use a singleton is to have easy access to a single resource. Since objects are passed by reference, a single instance can be shared with multiple objects naturally. From there, it's about good design and delegation.