Jenkin’s is being used to talk about the whole family, so they became one unit or object. It’s as if you are talking about one person. So in this case, Jenkins’ makes sense. Jenkinses’ is not correct since once the family is grouped together, they become a single object.

This is quite a big question about quite a small point really haha. If i could refer you to Wikipedia here, they say there has been much debate about this but modern thinking would suggest: Jenkins'es However in Wiki it claims that this is ridiculous and Jenkins' is much more sensible. My advice would be to use whatever you feel is comfortable and it's VERY unlikely that anybody will criticise you hehe!! Note though that in speech it would probably be pronounced as "Jen-kins-es" 3 syllables. Does that help or hinder?

If the surname ends in 's' (Jenkins), you should write "the Jenkins' Dentist". In practice, the apostrophe is often dropped and many would write "the Jenkins Dentist". The term "the Jenkinses" (to describe the whole family of Jenkins) is very informal and mostly out of date. It sounds odd to hear it now and the listener may wonder if you were being a little impolite and poking fun at them (the Jenkins family).