Q:

Meaning of laces in this verse

From Tim Minchin's GroundHog Day Musical:

There will be mornings
You'll be utterly defeated by your laces
Days when every look looks condescending
  Empty smiles in empty faces
  The same old places this stunning stasis 
  [...]

The only meanings both I and my dictionary know for laces are 

the shoe laces you have to tie
the lace as in lace dress

However, I can't make any sense out of the verse with either of the two meanings.
The only thing I can imagine is that it's somewhat metaphorically referring to when you're so tired in the morning you can't tie your shoe laces and thus get "defeated" by them 

A:

Laces refers to shoelaces in this song.  Tying your shoelaces is commonly used as a stereotypical example of a very easy task.  
So this lyric...

There will be mornings 
You'll be utterly defeated by your laces

... is trying to say you will have mornings that are so bad, you won't be able to do the easiest of all tasks, tying your shoelaces.  As you said, perhaps you are too tired, or perhaps you're just too frustrated and emotional so your fingers just won't move properly. 

Examples of "can't even tie their shoelaces" used to imply a person is incapable:
"Well I know Goldman, like the back of my hand, and I have serious doubts that he can even tie his own shoelaces unaided..." (from The Gold Pig)
Jason Derulo on getting injured, "You can't do normal things, man... you can't tie your shoelaces... the simplest tasks were suddenly the hardest things in the world".
Article title: Tech-savvy kids can use an iPad but can't tie their own shoelaces
Article title: The FBI Accused Him of Terrorism [But] He Couldn't [Even] Tie His Shoes.

