I have only 1 card, credit is "good." All paid off. The customer service is progressively worse, to the point that I want to get rid of it.

Smart thing to do would be to get another card first, then cancel this one.

In case I do that first, and in the case that I don't, how much would it hurt my credit to close this card?

I have 0 total balances across everything, so my utilization rate doesn't actually get worse if I just close this first (0/x still = 0).

My available credit would go to zero, I guess that's very bad? Just couldn't quite wrap my head around that fact (it's more "I'm not a good investment, because I barely use credit, thus score goes down," more than it is, "risky/too high balances/utilization, thus score goes down).

Doubt it impacts your score too much. But you will be losing out on the shelf life of that card which will result into much lower or younger "average age of accts" if u are to open new lines in the future.

You can run simulations online to see how closing it will effect your score. I am on my phone and cant dig up a link

If you indeed do have good credit you should get a different card. Consider looking for one with a decent cash back program and running your monthly expenses through it. Free money is nice.

I think having 0 balances will be pretty bad for you if you want to acquire more credit. It's kind of silly but having access to money and not using it looks good, not having access to money and not using it is a lot less impressive.

Also a major component of credit is avg account age. Keep that in mind as you close your only account. Honestly you should probably just keep it and get a second account somewhere else.

How long have you had this card?
What about the CS is bad if you don't use it?
What is your credit score currently?

Do you have an annual fee with that one card? If there's no fee, then you should just leave it open to build history over time.

Certainly do not close the account before opening up anything else. Having no credit is generally worse than having bad credit.

Open a card with no fee if current card has fee. Having no credit is almost as bad as having delinquent accounts.

Most of your gains in your score you got from this card will go away. I had a 800+ score at age 23 before my parents took me off their card.

Cancelling your only credit card will hurt your credit score. Unless your annual fee is more than say $60 it's good to have. What if you need an uber or rental car? Hotel room?