I often hear the word notoriety used in place of the word fame. The two are not synonymous. One has a positive connotation, the other a negative one.
I once heard a Southern Gospel singer use notoriety as he spoke of his growing fame. No, Mr. Singer, you do NOT want notoriety. Why? Because although notoriety indicates fame, it is fame for a bad reason. Notoriety has a negative connotation.
Notoriety, according to my trusty Webster's, means a being notorious (don't you hate it when a they define a word by using the word being defined?) and notorious means well known or commonly known, especially for a bad reason; widely but unfavorably known or talked about.
So unless you aspire to be well known for the wrong reasons, don't say you desire notoriety.
Coming next month in Grammar Goofs: My issues with a common Christmas phrase.