As BB says, initial screening is done with PLM on stained specimen. TEM is used for follow-up.
Sampling is reasonably important, as is knowing which materials to suspect...such as 9"x9" composite floor tile as compared to 12"x12" floor tile. Most of the 9" tile contains asbestos, while some of the 12" tile contains it. Neither is generally friable so cleanup is a bit easier. Many wall and ceiling plasters (lime or gypsum based plasters) contained asbestos. These are usually friable.
Asphalt based adhesives for floor tile usually contained asbestos fillers. Asphalt flashing cement produced before about 1995 contains asbestos as a filler. Many roofing felts, both general felts and flashing felts produced before about 1990 contained asbestos. Many pipe insulations contained asbestos and "popcorn" ceilings popular in the 60's, 70's, and 80's contained asbestos. Almost any vermiculite was contaminated with asbestos fibers.
Outside the production of asbestos containing materials (ACM), I don't know that there has ever been a documented health issue with incidental exposure to ACM. It was a big scare for public buildings, particularly schools in the 1980's and 1990's.