As we approach Rosh HaShanah, it is essential to be aware of the halachically permissible times to recite Selichos (Prayers asking HaShem to forgive us), as well as point out somewhat "commonly accepted" times to recite Selichos which Jewish law forbids or strongly discourages. The following is taken from a halachic ruling by Rav Moshe Feinstein, zt"l, relating to whether one may recite Selichos before chatzos. Chatzos refers to halachic midnight. Chatzos is the precise midpoint of the night. When calculating chatzos during the daytime, we take the midpoint between sunrise and sunset. If the daytime chatzos is at 12:45 pm, the nighttime chatzos would be at 12:45 am. It is perfectly fine to have a post halachic midnight Selichos, however to have a post-midnight Selichos (12:00 am) is problematic, no matter how many people decide it to be "an accepted custom".
Rav Moshe Feinstein, zt"l on When to Say Selichos: Igros Moshe: Orach Chaim (section 2), Siman 105
In response to the question whether Selichos may be recited prior to chatzos, Rav Moshe Feinstein, zt"l, ruled as follows:
Rav Moshe on reciting Selichos during the last third of the night/the end of the night:
Last third of the night:We learn (I believe in maseches Avodah Zarah) that the last third of the night is the best time to recite Selichos as HaShem's Divine Presence floats though this world at that time. Selichos should be recited at a "favorable hour", therefore, when HaShem's Presence "floats around" this world, it is a time of Divine favor and therefore is a recommended time for Selichos. Rav Moshe quotes the Machatzis HaShekel as defining the night as ranging from nightfall (time when three mid-sized stars are viewable) until the first rays of dawn. The RaMBa"M rules that one should wake up shortly before daylight in order to conclude reciting Selichos shortly before it starts becoming light outside.
Chatzos: The gemara notes that chatzos is also considered a time of Divine favor and therefore many communities recite Selichos after chatzos.
First third of the night: The Rosh and the Tur rule that one may recite Selichos one-third of the way into the night as at each third of the night (each ashmura), it is a time of Divine favor (i.e. the conclusion of the first-third and second-thirds of the night are times of favor). When they rule that Selichos may be recited a third of the way through the night, they refer to at one-third of the way through the night, not after. Therefore this ruling does not permit 12 am Selichos.
Halachic Problems With Reciting Selichos Prior to Chatzos on Saturday Night:
The Sha'arei Teshuvah (Siman 581) rules that it is forbidden to recite Selichos and Vidui (confession of sins) prior to chatzos due to the holiness of Shabbos. As Shabbos recently concluded, we should not recite Selichos until after chatzos. In the siddur of the Rivatz, he notes that the Ari z"l did not permit mourning for the destruction of Yerushalayim on Saturday night. Rav Moshe notes that the Ari z"l must have meant that mourning for Yerushalayim was forbidden after Shabbos until chatzos, for that is the time when people may begin reciting Selichos on Saturday night. Rav Moshe notes that only in a type of situation where it would be impossible to recite Selichos after chatzos, one may recite Selichos earlier, provided that no one Selichos would not be recited; otherwise, there would not be an awakening to repent. If a community would fulfill the aforementioned criteria, they would have to announce that an exception is being made this year to recite Selichos earlier due to the pressing situation where Selichos would not be recited otherwise, thereby making early Selichos a necessity. In addition, they should inform everyone that next year the Selichos after Shabbos will be recited after chatzos.
My Addition: Morning Selichos: Many people find it difficult (possibly even unhealthy) to recite Selichos late at night. Often, those people recite Selichos in the morning, prior to Shacharis, (close to the conclusion of the night). When one recites Selichos during the night, he must make sure to not recite the phrases that would indicate that he is reciting Selichos at night. So, if one's choice is to recite Selichos at 12 am midnight (if it is not after chatzos) or the morning, recite it in the morning. 12 am is an arbitrary time which generally does not occur at either third of the night nor at chatzos.
Everything else being equal, I would recommend the following times to recite Selichos (in order of most recommended to least recommended):
1) Recite Selichos at the last third of the night and finish Selichos before it gets light outside.
2) Recite Selichos at chatzos.
3) On all nights except Saturday night, after a third of the night has passed.
4) In the morning.
5) Do not recite Selichos on Saturday night before chatzos - Beware of those 12 am minyanim.
With that, I conclude and wish all of Bnei Yisrael a successful Selichos and a happy and healthy year. K'siva V'Chasimah Tovah!