I have a 15-20lb wood bathroom cabinet (24 inches wide, few feet tall) and I'll be looking to load it up with heavy bathroom supplies. Problem is there is only one stud available. Do you think one stud + drywall anchors on the other side is enough support? I was thinking it'd be best to cut out the drywall and put a backer board in between the studs?
While having all four mount points connect to structure (aka: the stud) is ideal, I think in your case, having two mount points in wood and two in a drywall anchor, you're going to be ok.
Consider this question: What is the weight capacity of a drywall screw?
One drywall screw CAN (not should) hold a lot of weight for its size. Also a properly installed drywall anchor can typically stand 70~ pounds of pull force (as in pulling out, not down).
With two WOOD (not drywall - wood) screws set into the available stud and 2 (or better, 4) drywall anchors, I think you're not likely to exceed the capacity. If your cabinet total weight when full is < 100 lbs you're probably not going to have a problem.