The most popular, and cheaper, for early detection of trisomy, techniques are amniocentesis and chorionic villus biopsy. Although these techniques are invasive and can cause abortion up to 5% of cases. Then there are the non-invasive techniques such as triple biochemical screening and non-invasive prenatal screening. The latter can detect peripheral maternal blood, and from the tenth week of pregnancy, the presence of trisomies 13, 18, 21; in addition to the detection of fetal sex and the presence of aneuploidy. Its only drawback is its high cost and the few places in the world where it is performed.
It is to explain either you are trying to diagnose trisomy before birth or after birth. Anyhow, Karyotyping is the best method to identify trisomy. However, sampling technique can differ and it also has effects on results. Sampling techniques used for pre-natal screening are Chorionic Villus Sampling and Amniocentesis.While, blood can be used in a living baby to diagnose condition.
You did not state which material you are working with. However, trisomy can be diagnosed using karyotype analysis. This applies to all materials or samples involved. However, the protocol used to effect the karyotype analysis may differ for several reasons.
Constitutional chromosome analysis or cancer chromosome analysis will have different implications on trisomy detection approaches. Most of the information is missing, however let me try to give a meaningful answer. If you know which chromosome trisomy you are suspecting, FISH using chromosome - specific alpha satellite DNA probe can be quicker and simpler as one does not have to wait for the metaphase chromosomes. If which chromosome trisomy one is looking for is not known - and even as a gold standard of cytogenetics the GTG banded karyotyping is recommended. I am not clear how Flow cytometry can be useful - it can detect gross aneuploidy, but trisomy of a particular chromosome am not clear - and how it will be better than FISH. Textbooks like AGT manual of cytogenetics are a must for any one working with the human chromosomes.