When you 'zero/zoom/home in on' something, you get closer to a goal or you focus your attention and your efforts towards achieving something or getting somewhere. You might 'zero in on' that promotion you've been looking for, or 'home in on' a bargain you saw from the opposite side of a busy store. 'Zoom in on' would be heard less, and might sometimes sound a little awkward, because it generally applies to fast physical movement or making something visually larger to examine it closely. In this regard, it's less figurative than the other two options.

1. When we went to the park yesterday, a swarm of bees zoomed in on Paul and stung him repeatedly.
2. As soon as the child entered the room, she homed in on the cupcake on the table and took a bite from it before anyone could stop her!

1. Rebecca zeroed in on the one mistake that I made in my report, and ignored all the totally valid points that I raised. 
2. If I may, I'd like to zoom in on the issue of the company's debts for a moment. Is there a way we can reduce the amount that we owe without cutting jobs?
3. I'm not surprised that David got that promotion. When he homes in on a goal, he doesn't let anything stand in his way.