Okay, so the serious answer... Your service dog can go with you to the doctor's office, because that is a place of public accommodation.
If you will be unable to handle your dog (say you're going to have serious work done at the dentist, or having an MRI or something) you cannot expect that the office staff will take care of your dog for you (that's not their job), so you should leave the dog home. If you're just going to the dentist for a routine check up, then the dog is probably okay. My dentist's staff is happy to take my dog out of the room while they shoot x-rays. I've actually never had any problems with taking him to longer tests, either (I've always called ahead to find out). I had an MRI that I had forgotten about, so I didn't leave my dog at home that day. So I stopped in about an hour before the appointment and asked them if arrangements could be made, or if I should drive him home and come back. They said that it was no problem, and that he could be locked in the dressing room. When I showed up for the appointment, the desk staff looked at him and said that there was no need for him to have to sit all alone in the dressing room, because they would be happy to have him hang with them at the desk. But this is a case of the staff going beyond the legal requirements. The best advice is to either leave the dog home or bring a back-up handler.
If you're going to be in the hospital, the best advice is again to leave the dog home. If you're in the hospital, you are sick enough that you probably won't be able to fully take care of and handle the dog. But if you are alert enough to handle the dog and can arrange for his care, then they generally can't exclude him. When I was in the hospital for three days, the nurses said my dog could come visit, so he did (and he even crawled up into bed with me, after the nurse said it was okay). But I'm really glad that I didn't have to worry about keeping track of him, feeding him, or toileting him while I was there. One of the hospitals in my area will let patients' pets visit them as long as they get the proper forms filled out by the doctor and the vet.
The allergist is sort of a gray area in the law. The DOJ does say that allergies are not generally a reason to exlude a service dog. The allergy would have to be so severe that it rises to the level of a disability (anaphylactic shock or similar). So if your allergist just ends up with a runny nose, then legally that's not a good enough reason to exclude the dog. I had an allergist before I moved that had no problem with me bringing my dog in. Then I moved and went to a new allergist once, and after that they called me and said that I would not be allowed to bring my dog into their building because they were worried about other patients having serious reactions. So they were arguing that the presence of the dog would be a fundamental alteration. It is possible that a court could decide that a dog in an allergist's office could be a fundamental alteration, but no court has looked at that issue yet, so nobody knows what the answer is.