Any one of those will work just fine. Do you plan to share a login between the forum and any other site functions? That might be easier in the same domain.

Heres how I recently setup my new Vbulletin:

www.mysite.com/forums = VB home
www.mysite.com/ = index page I created in dreamweaver.

Installed vbSEO mod (it's great! dont use VB without it)

setup robots.txt like this:
User-agent: *
Disallow: /forums/ajax.php
Disallow: /forums/attachment.php
Disallow: /forums/calendar.php
Disallow: /forums/cron.php
Disallow: /forums/editpost.php
Disallow: /forums/global.php
Disallow: /forums/image.php
Disallow: /forums/inlinemod.php
Disallow: /forums/joinrequests.php
Disallow: /forums/login.php
Disallow: /forums/member.php
Disallow: /forums/memberlist.php
Disallow: /forums/misc.php
Disallow: /forums/moderator.php
Disallow: /forums/newattachment.php
Disallow: /forums/newreply.php
Disallow: /forums/newthread.php
Disallow: /forums/online.php
Disallow: /forums/poll.php
Disallow: /forums/postings.php
Disallow: /forums/printthread.php
Disallow: /forums/private.php
Disallow: /forums/profile.php
Disallow: /forums/register.php
Disallow: /forums/report.php
Disallow: /forums/reputation.php
Disallow: /forums/search.php
Disallow: /forums/sendmessage.php
Disallow: /forums/showgroups.php
Disallow: /forums/subscription.php
Disallow: /forums/threadrate.php
Disallow: /forums/usercp.php
Disallow: /forums/usernote.php

Setup .htaccess with the typical non-www to www 301 (mod-rewrite)
and index.html to ./
(sorry don't have access to that on this comp, but the code is posted over and over here at ww search "duplicate content" to find it)

Then I spent several hours at vbulletin.org reading posts. There are a few mods there I will likely install.

I am still in the learning/testing phase of VB, I am a phpbb veteran, and I must say I am VERY impressed with VB so far. Already starting to build a community, the new board is less then a month old and it already has conversations going on.

My old board is 10 years old and in full swing, I hate to disrupt it by converting to VB from phpbb so I will likely leave it phpbb, but if I do anymore boards it will likely be VB from this point on.

I don't know why I do this but I always have when adding or including a forum...

www.mysite.com/forum/

but open it up in a new browser window so that the main site remains underneath... also offering the option to bookmark directly to the forum...

In a sense it keeps them apart while still being together...

Thanks for the information! Especially the seo mod!

I won't be sharing any login information between the two sites.

I have my primary website on a windows server with active server pages, so I don't have the htaccess functionality. I know, I could use something else, but I think I'll go for the forums.mysite.com or just mynewsite.com being the forums.

If you decide to make the homepage of the web site the "main" page of the forums, make sure you edit it extensively. There are a TON of vb Boards out there, and with little to no modifications to the home page you will get hit with duplicate content.

You can do it successfully this way, just make sure you add a lot of text to make it unique.

I'd avoid using "forum" or "forums" in the URL (even though it's user-friendly), as that is a magnet for spammers, hackers, etc. On sites where I have no forum I'm constantly getting 404s for hits to a nonexistent /forum/ directory.