Has the FIDE rule against bringing the game into disrepute been ever invoked? Indeed. I have found several examples of applications of the rule, based on the analysis of many resolutions by FIDE, and a few national and regional chess federations.
N.B. Although I tried to find references in English, it was hard to do and most of the references I found are in French, Catalan and Italian. The references I could find depend on my knowledge of languages and on the publication of disciplinary resolutions by a particular federation. (For example, I could not find such type of documents neither at the USCF site nor at the English Chess Federation's).
Cases of application
I actually found a case similar to that presented by @Brian Towers for committing illegal moves/actions during a game of chess. During a rapid chess competition, the white player repeatedly made illegal actions during the last two moves of the game (captured a pawn but did not take it off the board during his own time; retired the pawn during the opponent's time and, because of this movement, displaced a bishop from d4 to f3 and interfered with the opponent; knocked down two pawns while making his next move and did not set them up during his own time). This went unnoticed by the arbiter but there was video footage. The black player made a claim, but lost by time because of this. The result of the game was overturned based on article 11.1 and 11.5 (not annoying the opponent) (decision 3/2016, in Catalan, by the competition judge of the Chess Federation of the Balearic Islands).
However, this type of application of the article seems to be rare and it is more often invoked to punish inappropriate behavior during a match. Here are some examples:
- A player broke havoc in disguise during the final of 86th French Championship as part of a prank. According to the arbiter that reported the event, "His behavior and actions have brought discredit to our sport". The player was suspended for two years (Resolution 2011-14 of the Federal Committee of Discipline of the French Chess Federation, in French).
- An adult player shouted to an underage player, accusing him of having touched a piece and not played it. The adult player was suspended for 30 days (Resolution 2020/277 of the National Sports Judge of the Italian Chess Federation, in Italian).
- Two players started quarreling during a match about some pieces having been knocked down. Both players lost the game and were suspended for 30 days (Resolution 2019/259 of the National Sports Judge of the Italian Chess Federation, in Italian).
- A player insisted on using swear words and acting disrespectfully, even after several requests by the arbiter. The player was suspended for two months (Resolution 2017/155 of the National Sports Judge of the Italian Chess Federation, in Italian).
- While playing against an underage player, an adult player repeatedly went to the bar for alcoholic drinks (although he apparently was not inebriated). He was reprimanded on the grounds of article 11.1 (Resolution 20/2019 of the Catalan Chess Federation, in Catalan).
- A player was reprimanded on the grounds or article 11.1 for shouting during a match when his opponent's phone alarm rang (Resolution 7/2019, of the Catalan Chess Federation, in Catalan). (The other player was reprimanded for telling the delegate, but not his opponent of the fact that the alarm would ring to remember him to take his medication. The competition rules specified he should have told both about it.)
- Several players were reprimanded for misbehaving (shouting, speaking aggressively) because of a conflict during a match (Resolution 07/2017 of the Catalan Chess Federation, in Catalan).
- A team's delegate was reprimanded for acting angrily when making a claim (Resolution 06/2017 of the Catalan Chess Federation, in Catalan).
- A player was reprimanded for refusing to shake hands as customary at the beginning of the game (Resolution 17/2016 of the Catalan Chess Federation, in Catalan).
- An adult player has suspended for six months for acting pejoratively against two underage opponents. The decision was confirmed during an appeal based on grounds of article 11.1 and 11.5 (Appeal Resolution 15/2016, of the Catalan Chess Federation, in Catalan).
Article 11.1 has also been used to punish affronts and offenses against chess organizations:
- The president of a chess club was suspended for three years for offenses against the French Chess Federation on Twitter and for manhandling a couple of underage players he trained (Resolution 2019-14 of the Federal Commission of Discipline of the French Chess Federation, in French). Although the resolution does not cite explicitly the article 11.1, it says that such actions (the insults on Twitter) "are of such a nature to attack the image of the French Chess Federation and the game of chess" (my translation).
As is apparent from the examples above, in case of minor transgressions penalties are limited to admonitions with more severe punishments in case of re-incident behavior. In more severe cases it results in the loss of the game or even license suspensions.
Alternatives to article 11.1
The invocation of article 11.1 seems to depend on the actual regulations of each federation. In fact, the offenses listed above are related to violations of ethical codes in some federations.
For example, the article 2.2.10 of the FIDE Code of Ethics is approximately equivalent to article 11.1 of the Laws of Chess. It states:
"In addition, disciplinary action in accordance with this Code of
Ethics will be taken in cases of occurrences which cause the game of
chess, FIDE or its federations to appear in an unjustifiable
unfavorable light and in this way damage its reputation."
This article was used to suspend former FIDE president K. Ilyumzhinov (case 2/2018 of the FIDE Ethics Commission) for pursuing his own interests instead of FIDE's in addition to other actions.
Also, in France disrespectful behavior is usually judged as violations of the Ethics and Deontology Chart of the French Chess Federation (e.g., Appeal decision 2017-02 for hitting a player, decision 2019-01 for insulting and menacing two players, decision 2019-13 for shouting to the opponent, in French).
Finally, the ruling 2/2020 of the FIDE Ethics & Disciplinary Commision regarding an attempt of match-fixing states that "Art. 2.2.10 is a more general rule, primarily aimed at targeting acts or omissions by players or office holders that is not covered by any of the more specified articles in the Code of Ethics." I think this should also be applicable to article 11.1 of the Laws of Chess.