A bookcase that rocks when a kid climbs it is a small problem with an easy fix. Most tall furniture comes with a way to anchor it to the wall, and the job is mostly finding the right spot and using the right hardware. The real question is whether that job belongs to you, to a free weekend, or to a hired hand.
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Check the furniture and the wall first
Look at the back of the piece and the instructions for the anchor points the maker shows. Those are the spots the hardware is meant for. Then look at the wall. A stud finder tells you where the framing sits. Find out whether the wall is drywall or plaster, because the anchors that work in one often do not hold in the other. Check each house the same way, since the two walls will not match.
Do it yourself when
You have a drill and the basic tools, the wall is standard drywall or the framing is easy to locate, the furniture has clear anchor points, and you can test the result by giving the piece a firm pull once it is fixed. Keep the room clear of little hands while you work, and have another adult nearby if you can.
Hire it out when
The wall is plaster and you do not know what sits behind it, the piece is too heavy to move alone, or the room has wiring or plumbing you cannot see. Hire it out too if the job is simply not something you want to spend a Saturday on. That is a fair reason, and it does not mean you are doing this wrong.
What to ask before you hire
Ask exactly what they will drill into, which hardware they will use, and whether they will patch the holes. Ask whether the work is covered if something gets damaged during the job. Ask for the scope in writing so both houses get the same job. Ask how long the visit should take, so you can plan the kids around it.
Once the piece is anchored, give it a quick check every few months, especially after a move or a new climbing phase. Tighten whatever has loosened, and write down when you checked so the next person knows when it was last done.