What cell structures are usually seen under the compound microscope?

It depends on the kind of cell, the staining technique (if any), and the power/resolution of the 'scope. With a typical clroom microscope, you'll be able to see a cell wall of plant or fungi cells, maybe the cell membrane (as a line at the edge of an animal cell), and the nucleus. If the plant cell is large and photosynthetic (like *Spirogyra* or *Elodea*), you can see the individual chloroplasts. You can sometimes make out the intracellular inclusions in a large, stained leukocyte.