hiragana - What are the right shapes of kana?

I have been learning Japanese for over a year now, and I have noticed that some people are writing kana differently. For example, I have seen う being written just like how it appears now (looks like a rotated U letter) and I have also seen it being written like a question mark (only a bit more curved)!! i

So I was wondering - what are the correct shapes? Or are they all correct and it's just a matter of rightly forming it?

Answer

The typeface used in textbooks is called 教科書体 (lit. "textbook typeface") and fonts are designed to be copied as closely as possible.

(This is quite different from languages based on the alphabet: there are many different cursive styles taught in schools and textbooks are never typeset in cursive "handwriting" font.)

That said, like you observed there are very slight variations. For う, there is a question of whether the first stroke should have a hook or not, and whether the second stroke should be entirely round or have a dent:

I would say the first one is more standard for learning handwriting; the second and third ones are shapes used in current 国語 and 算数 textbooks.

Any of these should be considered "correct", but even when aiming for any of these, actual handwriting will look different again.

For reference, here are two charts you can follow:

(from happygolucky.info (PDF))

(From this blog (image link))

Finally, there is a very nice instructional video for learning to write ひらがな (with a calligraphy/brush pen, but it also applies to pens/pencils) that I learned from @snailplane: