2 difficult leniencies in halacha
There are 2 kulos in halacha that are very widely adopted and are very difficult.
1. Chodosh is permitted in חוץ לארץ
2. The fact that you need 600,000 people to make a רשות הרבים
I thought about these now because Pesach just passed meaning that there is no problem of chodosh, and the daf yomi just started learning the first perek of shabbos where the issur of הוצאה is discussed
1. There is a Torah prohibition not to eat any products of the five types of grains specified in the Torah (wheat, barley, oats, spelt, and rye) before the second day of Pesach, the 16th of Nissan, when the Korban Omer was brought. If planted before the 16th of Nissan, it may be used after that time. However, if planted (or actually take root) after that time, it may not be eaten before the 16th of Nissan of the following year. Therefore, wheat etc., planted during the summer and harvested in the fall, may not be eaten until after the next 16th of Nissan, in the spring. During the time that it is prohibited, it is known as "Chodosh", the new grain. When it is permissible, it is known as "Yoshon", the old grain. All grains sold during the spring and the summer can be assumed to be Yoshon, since it is likely that they were planted before the 16th of Nissan. The problem starts to arise when the spring and summer grains hit the market in the fall.
According to all the standard rules of pesak we should pasken that it applies in Chutz laaretz as this is the position of most of the Rishonim (including the Rambam). Everyone relies on the Bach who was matir. In his day and throughout the generations in Europe (as the acharonim say) it was a question of survival, without chodosh Jews would have had a hard time surviving. Obviously today the situation is much different.
The reason why this only recently became a problem in America is because for many, many years America had a surplus of wheat that they would store, and would only sell the older wheat to the millers. Therefore, all wheat was Yoshon. However, approximately 10-15 years ago they sold off their surplus wheat to the Russians, and the new wheat is hitting the market before the 16th day of Nissan. Although they once again have a surplus, the experts say that they are storing the older wheat indefinitely, and allowing the new wheat to come to the market.
Note: in Israel this is not an issue as chodosh is prohibited according to everyone and therefore all the hechsherim are makpid on chodosh.
2. One of the melachos of shabbos is הוצאה, you are not allowed to carry an object from a private domain to a public domain, in addition you cannot carry an object 4 amos in a public domain. The gemara at the beginning of Shabbos (daf yomi), define what is a רשות הרבים. Nowhere does it say that you need 600,000 people. Rashi (Tosafos also) came up with this idea in Eruvin. It is very difficult for a number of reasons, chief among them it doesn't say it in the gemara.
Why is this important? All our eruvin rely on the fact that we don't have a רשות הרבים min hatorah because they use a צורת הפתח (this requires a whole separate discussion which is not for now). Why don't we have a רשות הרבים min hatorah? because we lack 600,000 people (this is a big part of the controversy of the Brooklyn and Manhattan eruv). If we deny this requirement of 600,000 people, then all our eruvin are no good. Again, the accepted principles in psak would dictate that we should follow the more stringent opinion, however the minhag is to be מיקל.
The truth is that by chodosh more and more people are starting to be machmir. If you go into most of the kosher pizza stores in Brooklyn or in Queens you will see a sign קמח ישן. This really requires the kashrus agencies to decide that they want to be makpid on this.
The issue with eruvin is more difficult because without an eruv you condemn the women with little babies to sit inside all shabbos. I know a number of people who they themselves are machmir not to use an eruv, but allow their wife to use the eruv. This is very difficult to me, if you think it is assur how can yu let your wife do it and if you think it is mutar so why are you machmir?
In any case, we find that in these 2 cases the minhag seems to override the halacha.