Anything that asks you to Update Your Info or you'll Lose Your Account, and then provides some funny looking link where you have to click and log in is clearly fraud.
Anything that states along the lines of "the credit card on file is about to expire, please log in to www.some website.com - with your user ID and password to update it", and then leaves a big long message about never sharing your password is generally safe enough. So long as you're going through the site's main page and not clicking anything special in your e-mail, it's all good.
One giant red flag for scams, which I realize not all online users can recognize, is mistakes in grammar and punctuation. Consider that many of these scams originate in Eastern Europe or Asia. The culprits are apparently not as literate in American English as they would like to think.
Oh, God. So many of those just crack me up, but I especially like the ones where they keep adding characters to the original picture; the giant hand that squashes your head like a grapefruit; the Robocop one and the Mortal Kombat one.