The Volokh Conspiracy
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The sticker appears to be a version of this (UPDATE: see the WKRN story, which I erroneously failed to link to at first):
But of course this is constitutionally protected, given the Court's decision in Cohen v. California (1971) that a jacket bearing the words "Fuck the Draft" was constitutionally protected. Indeed, it's not even barred by the relevant Tennessee statute (Tenn. Code § 55-8-187),
To avoid distracting other drivers and thereby reduce the likelihood of accidents arising from lack of attention or concentration, the display of obscene and patently offensive movies, bumper stickers, window signs or other markings on or in a motor vehicle that are visible to other drivers is prohibited and display of such materials shall subject the owner of the vehicle on which they are displayed, upon conviction, to a fine of not less than two dollars ($2.00) nor more than fifty dollars ($50.00). "Obscene" or "patently offensive" has the meaning specified in § 39-17-901 [which restates the First Amendment test for obscenity, and defines "patently offensive" as "that which goes substantially beyond customary limits of candor in describing or representing such matters," apparently referring to sexual matters.
The word "fuck" in this context doesn't fall within the First Amendment obscenity test, as the Court recognized in Cohen:
This is not … an obscenity case. Whatever else may be necessary to give rise to the States' broader power to prohibit obscene expression, such expression must be, in some significant way, erotic. It cannot plausibly be maintained that this vulgar allusion to the Selective Service System would conjure up such psychic stimulation in anyone likely to be confronted with Cohen's crudely defaced jacket.
The same logic applies here; indeed, the Tennessee Attorney General's office has acknowledged (Tenn. Op. Atty. Gen. No. 88-44) that
The [statute] will not reach bumper stickers that are in extremely poor taste but are not obscene [under the Supreme Court's obscenity precedents]. For example, bumper stickers such as "s..t happens," although unquestionably in poor taste, do not meet the constitutional or statutory standards for obscenity because they do not appeal to the purient interest. Consequently, they cannot be banned as obscene.
See also Cunningham v. State (Ga. 1991) ("Shit Happens" bumper sticker constitutionally protected).