One of the most famous examples I have heard is that one should not even tie one's shoes like the non-Jews do in times of persecution. That seems rather excessive. What if the Jews started doing some custom, and it gets co-opted by the non-Jews. Can we still do it? If not, that could wipe out a lot of Passover seder traditions, since a large fraction of Christian (in particular Catholic) ritual is based on the last supper, which of course was a seder.
The best treatment of the topic I know of is a responsum by Rabbi Moshe Feinstein regarding a Polish Jew who moves to America and would like to adopt "American" clothing.
Rabbi Feinstein says we follow the opinion of the Ran, that chukat goyim is only something that the Jews haven't done until now, AND it has something to do with paganism or sexual immodesty. (Halloween is generally understood as an example of the former; for the latter, R' Moshe suggests that theoretically, if the Jewish women in a society clearly had the practice not to wear red clothing, then choosing to do so in imitation of the non-Jews -- red is flashier and more attractive -- would be a problem.)
Another possibility of chukat goyim is where the action makes absolutely no sense whatsoever, so the only reason to do it would be to try to blend in with non-Jews. A doctor's white coat (or the specialized clothing of craft guilds in Renaissance Italy) is okay because it's not pagan nor immodest, and you're wearing it for professional reasons. R' Moshe also writes that all the funny things about American clothing -- well, American men's clothing in the 1950s -- are considered decorations, and not a problem.
Fascinatingly, R' Moshe concludes that "American clothing" today is in fact "American Jewish clothing", no different than "Polish Jewish clothing" and thus not "goyish" at all. (Provided it's appropriate!)
The shoelace case is one where clearly the Jews had one shoelace style, and the non-Jews another. One choosing to change shoelace style would be making a statement "I want to look like a non-Jew." In this case, the non-Jewish style was also flashier, so the switch is both a statement, and a drift away from modesty. It's also important to realize that in times of persecution, we need to hold strong.
The laws of Chukkas HaGoy are discussed in the Shulchan Aruch, Yoreh Deah 178
The Rema (ibid. Seif 1) explains that for Chukkas HaGoy to apply then something must be done by non-Jews either due to pritzus (immodesty) or as an unexplained custom (since we are concerned it had idolatrous origins).
Rav Moshe Feinstein in Igros Moshe Y.D. 1:81 explains that for Chukkas HaGoy to apply Jews must be in the habit to refrain from the practice [at least when it comes to immodest dress]. Nevertheless he suggest that with respect to clothes that are really immodest that it is a possibility that they are prohibited due to Chukkas HaGoy (they are certainly prohibited due to their immodesty in any event).
It is the opinion of the Chochmas Adam 89:1 that when non-Jews establish a practice in the service of their religion then it is forbidden due to Chukkas HaGoy even if it is mentioned in the Torah that Jews had a similar practice (It would seem obvious that this doesn't apply to that which Jews are obligated to preform, I think our practice to avoid prostrating may be an example of this). If it is done by the non-Jews for no "good" reason then if it is mentioned in the Torah that Jews did so then they may continue so, but otherwise it is forbidden due to Chukkas HaGoy. The example given is there was once a practice to decorate Shuls with trees on Hag HaShavuos but the Gra nullified the practice because of Chukkas HaGoy. I suspect that l'maaseh it might not be as clear cut, but it seems like a very reasonable distinction to apply when faced with an issue. I would be interested if seen how other poskim address this question.