Q:

Can I cast Darkness on my shield?

The PHB says that 

A shield is made from wood or metal and is carried in one hand.
  Wielding a shield increases your Armor Class by 2. You can benefit from only one shield at a time.

Emphasis on "carried" and "wielding", because the Darkness spell description says that:

If the point you choose is on an object you are holding or one that
  isn't being worn or carried, the darkness emanates from the object and
  moves with it. Completely covering the source of the darkness with an
  opaque object, such as a bowl or a helm, blocks the darkness.

This left me a little confused as wielding and carrying seems to be two different things for the spell. 
Another interpretation of the spell could be that you can't cast it on something that another creature is wearing or carrying, to avoid situations like "I cast it on his/her weapon/shield/armor" and just leaving the creature on darkness without it knowing where/what the source is. 
If I can't cast it on my shield, is there any way to cast it on my weapon without wasting an action as the arcane focus occupies my other hand?
I've seen this answer, I could try to convince my DM to sell me a rapier that doubles as an arcane focus, but I doubt that he'll allow it.

A:

Yes
In the absence of specific game language, we default to regular English.
You are doing all three things to the shield: you are holding it, you are carrying it, and you are wielding it. None of those things are mutually exclusive, and in fact wielding it requires you to be doing the other two. On the other hand, one could be holding or carrying a shield without wielding it, by carrying it on their back or something. In other words, "holding" something is a broad category that contains "wielding" something. 
As you point out, the wording of the spell is clear that you can cast it on things that you are holding, and not things that other creatures are holding.

