After having seen a few pictures around the net of vintage restoration projects that showed what was quite likely asbestos being removed. and seeing a few odd posts about it here, I though I might post a bit of general info about it.
I have been an AHERA certified building inspector, and have worked in asbestos abatement. (I don't work in the field any more.)
The kinds of asbestos one might expect to encounter in a renovation project would be in flooring, and "thermal systems insulation" TSI for short.
As far as flooring the two main types are going to be vinyl tiles and vinyl sheet flooring. The sheet flooring, aka linoleum, can have a backing that ranges from 40 to about about 100% asbestos. it looks kind of like thick paper. This is the bad stuff. The glue adhered to this backing and when you pull the flooring off you are essentially tearing the asbestos in half. This means it gets airborne in a big way. Floor tiles (often, but not always the 9x9 ones) contain up to about 40% asbestos often about 15-20% the asbestos was bound up in the vinyl and only becomes airborne when the tiles is scraped or broken. The glue used to hold these things down can have asbestos in it also.
Thermal system insulation can be gaskets, and sheets of asbestos paper. These types of asbestos are quite easily made airborne.This is know as "friable" and is more hazardous then non friable like the vinyl tiles.
Any old heater or furnace is probably going to contain some asbestos. If it looks like white cardboard of paper, you probably have asbestos. Don't think because you have a more recent trailer it does not have asbestos, there are arguments about cut off dates. I did a test of flooring on a house from 1979 and it was positive.
A REGULAR DUST MASK WILL NOT WORK!
Asbestos fibers a super small. they will go right through a regular dust mask, dustmasks also have a bad seal against your face.
A shop vac, even with a "hepa" filter will serve to get the asbestos airboren and spread it around everywhere. It will also contaminate your shop vac.
Testing is fairly easy and cheap. Find a testing lab in your area. Preferably one that has 3rd party certifications like NVLAP and they will have instructions for removing a small piece and sending it in cost is about $25-35 per sample. Results come in 24 hours or so. If you have white, greyish, or tanish or any color really papery cardboard looking looking stuff in or around a heat source it is probably asbestos. Google images will provide quite a bit of guidance. Here is a good example of sheet vinyl.
Hope this helps