Twice now, I've had pus-filled spot on gum, which quickly burst causing pain/sensitivity. 3 dentists, 2 x-rays showed no infection. Any idea of cause?

Parulis fistula. is a sign that there is chronic painless infection and your body is draining it out. They do not show up always on an x-ray. Unlike some other types of infection, a chronic gum abscess will not get better on its own. See a periodontist or endodontist for definitive diagnosis and treatment. Good luck.
Endodontist. See a root canal specialist. Radiographs don't show everything. Was the tooth tested for vitality or probed for periodontal pockets? Your post did not say so I assume the answer is no.
Many variables. Did you go to a general dentist or a specialist? If you saw a specialist, did you go to an endodontist or a periodontist? There are many variables. Did the general dentist do a six point periodontal charting around the tooth? A fracture or periodontal abscess may be subliminal and not show up on the radiograph. Was an antibiotic given and did it alleviate any symptoms?