Could you tell me if plain oral X-ray is sufficient to detect if the upper palate has cracked during the wisdom tooth extraction? If not, which test?

A palatial x-ray . Also called an occlusal x-ray may be used. Typically oral surgeons have these films since they are used for trauma. A better choice is a 3-d digital x-ray called a cbct. Many oral surgeons and some general dentists like me have this technology.
A little. If an upper wisdom tooth in being removed caused damage, it is unlikely to be the palate. The palate is soft tissue at that area, and the tooth is commonly displaced toward the cheek in any case. So, if bony damage is done, it likely the tuberosity of the maxilla and and an x-ray may help, but palpation or moving the area with fingers will be the ultimate sign.

Related Questions

What is the best test with the least radiation to see if the bone near #1 tooth cracked during the wisdom tooth extraction due to lousy oral surgeons?

Digital x-rays. Digital x-rays use an extremely low dosage of x-ray and are very detailed. Teeth and bone breaking during removal of any tooth, and especially wisdom teeth, is common and most often not because of "lousy oral surgeons"'. Read more...
Your dentist. Your dentist should be the one to make that call. If you don't trust him/her, go to a specialist (oral surgeon) and they'll help you out. Sometimes just a visual inspection is all they need. The extreme is a cbct or "cat scan". Read more...
CBCT. A cone beam 3-d image would be a good option for diagnosing a crack in any surrounding bone. The bone is this area is often very thin and soft. It has nothing to do with a surgeon being 'lousy' or not. Read more...
Cone Beam CT. If your wisdom tooth was impacted, removal of bone may have been required, which some people may refer to a the bone being "cracked". Some erupted wisdom teeth may have very large wide spread roots with extremely thin bone over the roots that can be attached to the roots when the tooth is removed. Some bone loss with the tooth removal does not necessarily mean your surgeon was bad. Read more...