New Forest District Council use hyphen's sometimes, and other times not.
The Advertiser and Times generally avoid the confusion, and just simply call the village 'Milford'.
So there you have it, just like the article below, I still have no idea which is correct!
-----------------------------------
UPDATE
A helpful reader has suggested that we check the Domesday Book.
The Domesday Book is a manuscript record of the Great Survey of much of England and parts of Wales completed in 1086 at the behest of King William the Conqueror.
Sadly, we had done that previously, only to discover that the Domesday Book recorded our village in its old name of Melleford.
Bob Braid, a knowledgeable local historian, has bought to our attention to a map published in 1896 which tells another story.
In the late 1880s, Colonel William Cornwallis-West, owner of Newlands Manor, had an original intention to develop a high-class seaside resort in order to exploit the property he owned with magnificent sea views on the top of Hordle Cliff.
He wanted to emulate what his friend, the Duke of Westminster, had done at Eastbourne. Plans included houses, a hotel, gardens, and a hydropathic establishment. There were also ambitions to try to install a railway station.
In preparation, new roads named after family and friends, e.g. Cornwallis Road, De la Warr Road, West Road, Whitby Road, Westminster Road and Pless Road were laid out on Hordle Cliff.
The name 'Milford-on-Sea' was to apply to his new seaside resort development on the cliff top, with the village being known as simply 'Milford'.
However, the plan never attracted the finance it needed and his vision was never realised.
Colonel William Cornwallis-West was declared bankrupt in 1916. He vowed to repay the debt and did so by selling the family estates in Milford on Sea, including Newlands Manor, and most of its heirlooms.
Historical Map Changes
Historical local maps held in the Frith Collection show that in 1850 and 1907 the village was known simply as Milford. However, a map in 1919 shows the village name now as Milford-on-Sea, with hyphens. An Ordnance Survey Map from 1940 shows the village name as Milford on Sea, with hyphens.
So, as always, clear as mud. The quest goes on!
----------------------------------
Milford on Sea: Hyphens or no hyphens?
I woke up this morning thinking; 'Who can I annoy today'. Now, my wife had yet to have her second cup of tea, so it would have been unwise to even consider annoying her this early. A quick pop outside, but no one to be seen, so a change of plan was needed.
Luckily, there was a subject that had perplexed me for some time, so I decided now was the time for some research on whether Milford on Sea should have hyphens in or not. (We only deal with the big issues here you know.)
First point of call was the welcome signs into the village. No hyphens, so this must be right.
To double check I took a look at the Parish Council website. The header had no hyphens, but the official address details did, so did a couple of mentions in the site’s text. So ‘1-1’ so far.
The Parish Council avenue of research has now opened another dilemma, as looking at the 'Village Crest' not only did it have hyphen's, but also two men in strange attire.

One seems to hold a scythe, the other wearing fisherman boots, - plus a ship, a crown and a flower shape. Could be interesting to understand the meaning, but my hyphen mission had to take priority and be completed.
Next, the Community Centre. No hyphens on their website, so ‘2-1’ to the ‘No Hyphens’.
The NHS site for the War Memorial Hospital had no hyphens, yet the Hospital Friends website were clearly in the hyphen camp.
My next investigation should prove telling, Hampshire County Council mentions our village by using both versions, so they have covered themselves well by ensuring they have the correct version somewhere. 3-3 now.
Time to hit the streets, and ask a few random people in the village. My first questionee said; ‘You can stick your hyphens where you like’. I wasn’t entirely convinced he was commenting on my actual question. The next just looked at me pitifully & enquired whether; ‘I didn’t have something more important to do’. This was not going well, however, it did make me think, and I concluded I hadn’t.
In my very weird world, I was convinced that people needed to know the result of my mission. I therefore decided to take a new tact by finding out a bit more about ‘hyphens in names’.
Not necessarily relevant, but I discovered that the longest hyphenated town name in the USA is Washington-on-the-Brazos in Texas. I considered contacting one of the inhabitants to see if they would comment on my quest, but I decided they might not have heard of Milford on Sea, and even if they had, I am not good with accents.
I also discovered that Sutton-under-Whitestoncliffe is situated in the North Yorkshire Moors, and claims to be the longest hyphenated place name in the UK. Again, I never called anyone there for the same geographic & accent reasons already mentioned.
Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch is of course longer, but they missed the chance of a record by not adding any hyphens.
A quick look at Wikipedia, and although Milford on Sea village is mentioned, I am told you can't guarantee the website is correct, so that was disqualified.
Ah, try the encyclopaedia. Oh no, haven't got one of those anymore.
I finally had a look at BBC News website, but there was no mention of Milford on Sea at all, so no help here then.
I then had a brainwave, when does the Oxford Dictionary believe it is correct to use a hyphen? I found they are not sure either. The first editors of the Concise Oxford Dictionary, wrote in their preface to the 1911 edition: ‘We have also to admit that after trying hard at an early stage to arrive at some principle that should teach us when to separate, when to hyphen, and when to unite the parts of compound words, we had to abandon the attempt as hopeless, and welter in the prevailing chaos.’, they conclude ‘Hyphenation in English is highly variable, and in many contexts, it really doesn't matter’.
My ‘in-depth’ research therefore left the results as an honourable draw with the ‘With Hyphens-3’ & ‘No Hyphens-3’.
So, which is the correct way to write our village name: Milford on Sea or Milford-on-Sea? It seems there is no right answer, so just do what you like!
Thinking my mission was over, someone has just asked me which is correct, Danes Stream or Danestream? - Argggh!
-----------------------------