During the 2018 HMSAware campaign we took questions from the public and formulated them into an FAQ (currently under revision), with an Anonymous Question on the Wednesday evening, taking a look at that perennial question of whether or not hypermobility related health problems are rare. It is reproduced below, with an update taking into account more recent UK based research.

Hypermobility syndromes awareness week, 2018

An #HMSAnonQ: Is hypermobility rare? 

HMSA Answer: Hypermobility is thought to affect around 20% of the population - which is a huge number of people. When hypermobility causes problems it becomes a hypermobility syndrome (HMS). HMSs are a group of conditions that have hypermobility as a feature. 
Here in Europe, a rare disease is classed as one which affects less than 5 in 10,000 people. 
 Rare HMSs include:
  •  Marfan  syndrome: approximately  1  in  every  10000-15000 people. 
  •  Osteogenesis Imperfecta: approximately of 1 in 20,000 newborns. 
  •  Stickler syndrome: approximately 1 in 7,500 to 9,000 newborns. 
  •  Pseudoxanthoma elasticum: Published reports estimate it to affect 1 in between 25,000 and 100,000 people.
As well as many other conditions.
All types of EDS (with the possible exception of hEDS) are classed as rare - some so rare they've only been diagnosed in one family, others, like classical EDS and vascular EDS are more common than others, but still fall very much within this definition of rare.
Prior to the publication of the EDS classifications in March 2017 there was considerable confusion between hypermobile EDS (previously EDS3) and Hypermobility Spectrum Disorder (previously called JHS, HMS or BJHS) and and controversy over whether they were in fact the same thing leading to little consistency of diagnosis.
For around ten years they were considered indistinguishable from each other and often appeared side by side on medical notes, and while it appears that HSD is not uncommon there isn't yet sufficient clinical evidence to state with any certainty whether either is rare as the criteria have only been in use since 2017. Both may turn out to be more common than previously thought - or to be rarer - right now, we just don't know.
--------------------------------- Update February 2020 ----------------------------------
More recently, research in the UK looked at the diagnostic codes used by GP Practices serving NHS Wales. The paper was published in November of last year and used the diagnostic codes for Ehlers-Danlos syndrome and Joint Hypermobility syndrome and combined them to give an idea of how many people in Wales have a diagnosis of EDS (including the rarer types, as they're not distinct in the data). 
You can read the full paper (title and authors below) here

Diagnosed prevalence of Ehlers-Danlos syndrome and hypermobility spectrum disorder in Wales, UK: a national electronic cohort study and case–
control comparison

Joanne C Demmler, Mark D Atkinson, Emma J Reinhold, Ernest Choy, Ronan A Lyons, Sinead T Brophy

So with #2020Hindsight, is hypermobility rare? Still "Yes, No and maybe!"

In conclusion, it appears that while Ehlers-Danlos syndromes and Hypermobility Spectrum Disorder as described by the 2017 criteria have not yet been proven rare or no, like other heritable disorders of the connective tissue, it is readily apparent on anecdotal evidence alone, that health problems associated with hypermobility are not. Not all hypermobility leads to health issues, not all health issues associated with hypermobility will have a significant impact (particularly if they're picked up early and appropriately managed), they may not all be due to a connective tissue disorder, but hypermobility issues as a whole? Not uncommon.

Essentially, we still don't know, and until we do - health care professionals need to be treating the patient and their symptoms, not the label on their file.