My DH handles standard and non routine appts. if for some reason I can't. He has handled the ped., the dentist, the ENT, urgent care, and the orthopedist, the same doctors I have. We both work full time outside the home though so we have always shared that duty once I went back to work (when both of my girls were four months old). When DD1 needed speech therapy and tubes surgery he attended the initial visit, the surgery, and then when she had twice weekly speech therapy class we each took 3 hours a week for six months while she completed the speech therapy.
Both my DH and I work for the state government and have monthly sick leave hours that roll over so DH has over two months of sick leave banked (plus about 7 weeks of vacation) and I have about a month of sick leave and much less vacation (as I have to take 60-90 minutes of vacation each time he is out of town for work because I don't get to work until 8:30 and have to leave by 4:30 at the latest as I have a 45-60 minute commute some afternoons). Our jobs are pretty flexible unless my DH is traveling. My job doesn't involve travel.