1. Sniff Test
The most obvious way to test for whether you are sensitive to something is by smelling it. This is probably how you came to find out you were sensitive in the first place. The plus side is that people with MCS often have a very good sense of smell, being able to spot a toxin from a considerable distance, and this can help you avoid offending particles. The drawback is that not all toxins have a scent, and not all scents are toxic. This method is obviously skewed towards those things with more of a smell, i.e. essential oils may register as worse than flame retardants because they are more noticeable, but they are not more toxic.
2. Sleep-Next-To-It Test
This test is recommend by experts in MCS. You can put something you are testing next to your bed at night and see how you sleep and how it affects you in general. This is probably the most widely used test for MCS sufferers. I find this test difficult because the item you are testing is likely quite small and it can be hard to tell if a large quantity would affect you. Since the effects of toxins are also cumulative, this is a fairly big drawback. I'm not sure how much sleeping pills affect this test. Another challenge with this test is you need a fairly "clean" toxic-free environment in which you are sleeping and it can be difficult to keep other factors stable. Remember that changes in the weather, stress, emotions, medications, foods, chemicals sprayed outside or other exposures could affect your sleep or overall health.
3. The Erik Johnson Test
Erik Johnson's test for toxins involves first decontaminating and getting out into a very clean tent or trailer, somewhere where the air is really good and where you have not brought any of your former belongings. You can then lie outside, placing an item for testing on a tarp and see how you feel. Reactions can be delayed for up to four hours. The best indicators are anxiety, depression and insomnia. Remember that the effects of toxins are cumulative - the last item in a series of testing might not be the only offender. After you have tested it outside you can bring things into your dwelling one item at a time to test them. If you are testing a house/new location you will have to sleep there for a few days to find out if it is affecting you.
4. Naturopathic Test
There are various forms of muscle testing done by naturopaths and osteopaths that are quite good. Here is a how-to video. If you doubt the rationale behind it I would say suspend your disbelief for a while and give it a try. Use items you know to be tolerable as a base. If you are very in tune with your intuition and your energy and are very aware of how items, people, places, and sounds affect your energy, this could be a good method for you. I like to close my eyes during this test and try to avoid having what your brain knows about the object affect your body's intuition.
...And Go by What you Know
It is good to try all of these methods as none of them are a definitive way of testing every substance. I am also a big advocate for learning about what toxins are in everyday items and materials so we can avoid them.
You need the knowledge to know which factors to isolate and test for, and you need the physical testing to find out which of those is actually a problem for your body. You might also find things you are physically sensitive to that you didn't intellectually know were toxic.
But I think if you know something contains a toxin and you can avoid a toxin in your food, personal care products, furniture, clothing, building materials, air, then do that!