giving a shit 

To care about something.

I've stopped giving a shit about Luke.

giving a shit by Calieee January 31, 2015

not giving a shit

Not giving a shit has to do with the level of care given. Not to be confused with "not giving a fuck" see that definition. Not giving a shit means that you literally don't care enough to take action.
For example, Dillon may not want to go to work today. If he doesn't give a fuck, he will still go, but be filled with apathy. If he doesn't give a shit, then he won't bother showing up.

The kitchen staff are not giving a shit anymore so they don't do their fucking cleaning duties.

not giving a shit by Sushi snob June 30, 2017

not giving a shit about the Third World 

Pretty self-explanatory card from Cards Against Humanity.

Simon: Hey, what does not giving a shit about the Third World mean?
Justin: Basically it means you don't care about the problems of people from the most poor nations, duh.

not giving a shit about the Third World by EvanC7 December 31, 2023

Giving half a shit 

Not caring about a subject at all and being straight forward about it

A guy is giving half a shit that his brother is going to tell on him

Giving half a shit by I give half a shit July 1, 2020