Look in the yellow pages under analytical laboratory. You want to do a polarized light microscopy (PLM) test. This will determine whether asbestos is present, the percentage, and type. Tests cost about $35.00 and can usually be completed in a few hours to days depending on lab backup.
For a sample, you will want to cut off a piece of the linoleum or vinyl. Be sure to include all layers, especially the backing and any mastic. Asbestos content varies, and my experience has been that sheet products produced in the 60s and 70s has about a 40% chance of containing up to 15% chrysotile asbestos in the backing. Normally 4 to 8%. I have never been able to visually distinguish asbestos containing from non-asbestos containing, and I used to administer a statewide EPA asbestos and hazardous pollutants program.