When using a backer board for installation of cabinets, usually, at 34", 54"and 84", what thickness and type of backer board should be used? The walls in this old building remodel is plaster and lathe board and the location of the studs would make using a backer board preferred. Thank you.
You have to remove the plaster & lathe to do "backer boards" as they go between the studs & are typically 2x4's or 2x6's. If you're leaving the plaster & lathe then you're talking more about a Ledger or Hanging Board that gets screwed to the studs & just sits on the face of the lathed plaster.
These are typically just 1x6's or 3/4" plywood strips & should be shimmed to straighten & flatten the wall for the cabinets. You'll need to fill the end cabinets' gaps if using this method or skin the cabinet sides & scribe those to the wall.
However, you really shouldn't need either & you should be able to hit 2 studs for each larger cabinet's top & bottom. Use Beefy Coarse Thread Deck or Wood Screws, not drywall screws. In the case of small 12" wide type cabinets, they will be screwed to the bigger cabinets & supported solely by them.
Molly bolts, Mushroom-Heads or Spreading Anchors can be used just to suck them against the wall. Screw everything loosely to start with & then shim the cabinets to perfection beside the screws before sinking them. Use an impact driver if possible, this will sink the screws better & avoid chewing up screw heads.