According to the VSEPR theory, which shape is possible for a molecule with the molecular formula of `AB_3` (where the number of total electron groups is unstated)?

By shape, I'll assume you mean the molecule's molecular geometry; that is, the arrangement of solely the atoms around the central atom, not the nonbonding electron pairs. We would expect the second element (`"B"`) to have one electron shy of a complete valence, because there are three of this element, and one of `"A"`. So, element `"B"` must either be a halogen or hydrogen, and element `"A"` must be a nonmetal (it's a molecule) in groups `13` or higher (groups `12` and lower are all metals, except for hydrogen, which is a possibility for `"B"`).

1.

Let's start with a group `13` element, specifically `"B"` (boron). The compound boron trifluoride is `"BF"_3`, so it is an `"AB"_3` molecule. We can predict it's molecular geometry by drawing its Lewis structure:


We can see that since there are no nonbonding electrons left on the central boron atom, and there are three bonding pairs, the molecular geometry of `"BF"_3` is trigonal planar.

2.

You've probably heard of the compound ammonia, `"NH"_3`. Ammonia is also an `"AB"_3` molecule, and its Lewis structure is


Since `"NH"_3` has one nonbonding pair of electrons about the central `"N"` atom, and three bonding pairs, `"NH"_3` has the molecular geometry trigonal pyramidal.

3.

It is possible to form interhalogen compounds; that is, a molecule consisting of only halogen atoms. An example of this is `"ClF"_3`, which is also an `"AB"_3` molecule. Its Lewis structure is


We can see that the central `"Cl"` atom has two pairs of nonbonding electrons, and three bonding pairs, which makes the molecular geometry of `"ClF"_3` T-shaped.