Is there a correct spelling of Catherine/Kathryn Howard?

No. During the Middle Ages/ Tudor period, it was quite acceptable to spell names and other words in a myriad of ways. This was before the idea that there should be firm established spelling for every word. There were few English dictionaries back then, or established linguistic academies setting forth hard and fast rules about these things. Though Kathryn herself wrote her name with a K...However, it is very unlikely that if someone wrote a letter addressing her as Catherine, she would have thought twice about it.

You must also remember that the learned/ruling classes all wrote and read in Latin...and written English was still in its nascent stages.

In those days, it didn't really matter. In fact, that didn't happen until perhaps 100 years ago.

I do genealogy as a hobby, and I've seen documents were a given name was spelled 3 different ways in the same document! Spelling tended to be phonetic - you can often see when a new registrar starts work, as the last name of the family will change for all the baptisms. And then change again when a new guy takes over.

We are fussed about spelling these days, but, they weren't back then.

English spelling was all over the place in the 16th century - there wasn't a standardised way to spell words or 'wordes'.....

so none of the different spellings is the most correct ..you can choose the one you like best

you can see a copy of her signature in the link ...but to be honest it`s pretty much illegible

Although the first (very limited) dictionaries begin to appear in the Tudor period, spelling was extremely erratic. Even Shakespeare has difficulty knowing how to write his own name. So long as it sounded right that was all that mattered - as not that many people could actually read. So you see all sorts of variations of spelling for Aragon, Howard and Parr.

Up to victorian times spelling was not standardised, so people just thought how the word sounded and spelt it like that.Names were often spelt in different ways