Is it appropriate to use "You're welcome," followed by my name in closing an email?

I replied to an email which mentioned at the end:

Thank you,

Signature

with:

You're welcome,

Signature

Is such a reply correct/appropriate/common?

No, that just sounds weird. The "thank you" as an email signature is sort of a generic thank you to close the email with. You are being thanked for your time, any help or information you might be giving them, etc. You would be better off just signing off with a "thank you" of your own:

Thank you,

Signature

If they specifically thanked you in the body of the email for doing something, you could possibly say "You're welcome".

Dear X,

Thank you for organising such a big event. It was a big success with our clients. I have some questions for you about blah blah blah and I was hoping you could get back to me.

Many thanks,

Signature

Dear Y,

You're welcome. In answer to your questions, blah blah blah.

Best wishes,

Signature

However, unless they are thanking you for something big, "No problem" would be better. For example if they just said "Thanks for the quick response", then "You're welcome" would not be appropriate. Ignoring it is also okay and possibly even preferable, as long as you add in a few polite words and thank yous of your own. This is because emails are sent with a whole message all in one go, and "You're welcome" is a verbal response meant for a back-and-forth conversation. If you ignored a thank you in person it would probably be rude, but in a letter or email you don't need to reply to every line as long as you remain courteous as a whole. For some reason "no problem" fits in better to the email format than "you're welcome".

There was a recent topic on the differences between "you're welcome" and "no problem", but little of it applies to written communications. The email format is quite different because it's more of a missive than a conversation (whereas text or online messaging is still a conversation, even though it's written). However, you might still find reading around the topic interesting/useful so I have included it.

In summary, a lot of the time it would be more normal to not respond to the thanks directly (especially if it's just a signature-thanks) and just sign off with a "Thank you" or "Many thanks" of your own.

It depends on context.

Thank you,

Signature

Implies that the reader is performing some task at your request.

You're welcome,

Signature

Implies that you have done something at the request of the reader.

Neither is a good idea for a general-purpose salutation (either "Regards" or "Sincerely" are traditional choices), but both can be correct if driven by the content of the message.