Many people do not pick their own specialist or do so without much thought. After all, how different could two specialists be? I believe it can be a very important decision.
People will have differing ideas about how to pick the right specialist. Most go by convenience: do they work near you? For some, they prefer a doctor who can speak a language other than English. These are great reasons but I would like to add another key deciding factor which is: Are they the best person for your specific referral? When you need to see a specialist, it means your care has exceeded the scope of what your general practitioner is comfortable managing. If you have heart disease they may refer you to a cardiologist, if you have a broken bone you may be referred to an orthopedic surgeon. Whilst it's true that most specialists in a field would be able to take you on as a patient, the reality of modern medicine is that best practices are constantly evolving and updating meaning that more and more specialists are choosing to become subspecialists.
For instance, a cancer doctor (medical oncologist) may choose to only see lung cancer. Naturally, lung cancer would become the sub-field of oncology they are most up to date and experienced in. If you choose to see a general oncologist, you may miss out on receiving treatment from the person who is truly most suitable for your situation. Though oncology is a unique example, sub-specialisation is occurring in many fields of medicine and surgery. Proactive patients will be rewarded by doing their research before sending off their referral.
Therefore, my recommendation is to find specialists by subspecialty. Of course, many doctors will be comfortable treating many presentations, however for the highest likelihood of success, I recommend finding the most suitable doctor by looking at subspecialty. You can find subspecialty by googling or using the find a subspecialist feature on this website. The other benefit is that sometimes diagnoses can be treated by varying specialists. So by searching by subspecialist, you can ensure you don't miss out on finding the best doctor for your needs.