What methods in laboratory do they use to identify which antibiotic may be of value in treating a particular?

It's called a culture and sensitivity test.

They do a culture & sensitivity test. In that, they swab culture plates which are covered with a medium with a sample from the patients urine, blood, or wound. They then put the culture plates aside to allow the organism to grow (if there is an infection). They check the plates after 48 hours to see if anything is there & can see under the microscope what type of bacteria is present. If nothing has grown, the C & S is reported as negative.If bacteria grows & is  identified, they then do a sensitivity test where different antibiotics are added to the culture to see which one kills the growth. This shows which drugs are resistant (won't kill) and which will work. Usually the entire report is ready in 72 hours. Sometimes like with MRSA there isn't a common drug that works & then they may try other drugs, not commonly used.Further testing takes much longer.