维护:Singleton vs Dependency Injection

I have a script comprising lots of classes. In the classes' functions numerous mysql operations are carried out. I want to use a single datebase handle throughout the script. Thus, the Singleton approach or dependency injection (DI) might do this job.

However, I am wondering if I am right that using Singleton is the more efficient way from a maintainance point of view:

Assuming that all my (parent) classes finally extend the singleton class. Thus, in order to change something with respect to obtaining the database handle, I only need to do amendments to the singleton class. Particularly, I do not have to make any thoughts when adding new classes or the like since the database handle will be always present by extending the respective class of the Singleton class.

In contrast, when using DI I must provide code for creating the database handle in every script which might serve as a starting point (and passing the handle to the instances created). E.g. in script1.php, script2.php, script3.php, etc. Thus, amendments which should be made to the way the database handle is obtained must be made in every script, script1.php, script2.php, script3.php, etc..

The former approach looks much nicer to me in this respect than the latter. Or do I oversee some important thing?