How to change tab stops for a single paragraph in Microsoft Word 2007?

"Word 2007 didn't introduce any new behavior regarding this. When you select or place the cursor inside a paragraph, any paragraph-wide formatting will be applied to that paragraph only, retaining the integrity of the rest of the document.

Now, what you are almost certainly experiencing is a problem with the automated update feature for styles. This is an old feature (as old as styles in Word) that, once enabled, means any formatting changes to a paragraph using that style, will update the style definitions, which results in the remaining paragraphs using this style to be updated.

Let's get over with the simplest of cases:
This isn't happening to you on paragraphs using the Normal Style.
On this case:

just right-click the style from the
Ribbon Menu you are using on the
afflicted paragraph and select
modify.
Untick the Automatically update check
box

Hopefully this is enough for you. If not, read on...

Some users have this problem with the Normal style. Which is very odd.
I'm actually unsure as to why that would happen. If you right-click the normal style and select modify, you will see you don't have that checkbox. This style cannot be set to update automatically. But with some people it does(!).

To fix this issue:

Open the Document you are working on.
Press ALT+F11. This will open up
the VB Editor.
Go to the View menu of the editor
and select Immediate Window. This
is where we are going to type our
command to disable automatic updating
of styles.
Type: ActiveDocument.Styles(wdStyleNormal).AutomaticallyUpdate = False
Hit ENTER
Still on the VB Editor, click the
File menu and choose to save your document.

The new document will behave as you want it to.

EDIT:
Forgot one last bit of information...

It's possible that when you open a new Word document when firing word.exe, the problem will resurface either on the same regular style, or the Normal style (whichever was your case). To fix this, you'll need to update your Empty Document Template. To that effect:

Fire up Word from the executable or
icon in your Start Menu (do not
fire it up from an existing document)
Make the changes suggested above
again, whatever your problem was
(either a regular style, or the
Normal style).
After that, Right-Click again the style on the
ribbon bar and select Modify
Make sure Add to Quick Style list
is checked and check New Documents
based on this template
Click OK
Exit Word

New documents will now behave."