Short answer: neither. In most cases the code you really want to use is 303.
For the long answer, first we need some background.
When getting a redirect code the client can (A) load the new location using the same request type or (B) it can overwrite it and use GET.
The HTTP 1.0 spec did not have 303 and 307, it only had 302, which mandated the (A) behavior. But in practice it was discovered that (A) led to a problem with submitted forms.
Say you have a contact form, the visitor fills it and submits it and the client gets a 302 to a page saying "thanks, we'll get back to you". The form was sent using POST so the thanks page is also loaded using POST. Now suppose the visitor hits reload; the request is resent the same way it was obtained the first time, which is with a POST (and the same payload in the body). End result: the form gets submitted twice (and once more for every reload). Even if the client asks the user for confirmation before doing that, it's still annoying in most cases.
This problem became so prevalent that client producers decided to override the spec and issue GET requests for the redirected location. Basically, it was an oversight in the HTTP 1.0 spec. What clients needed most was a 303 (and behavior (B) above), but instead they only got 302 (and (A)).
If HTTP 1.0 would have offered both 302 and 303 there would have been no problem. But it didn't, so it resulted in a 302 which nobody used correctly. So HTTP 1.1 added 303 (badly needed) but also decided to add 307, which is technically identical to 302, but is a sort of "explicit 302"; it says "yeah, I know the issues surrounding 302, I know what I'm doing, give me behavior (A)".
Now, back to our question. You see now why in most cases you will want 303.
Cases where you want to preserve the request type are very rare. And if you do find yourself such a case, the answer is simple: use 302. Either the client speaks HTTP 1.0, in which case it can't understand 307; or it speaks HTTP 1.1, which means it has no reason to preserve the rebelious behavior of old clients ie. it implements 302 correctly, so use it!
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on the resulting page may be no bad idea. – Aetolia