Brachot 52 - Laws of the meal, Beit Shammai vs Beit Hillel
Some of the disagreements between the school of Shammai and the school of Hillel concern specifically the order of meals. For example, when one recites Kiddush (Sanctification) welcoming Shabbat and the Shabbat meal, then Beit Shammai says that he should first mention the day (Shabbat) and then the wine, and Beit Hillel reverse the order. Why? Beit Shammai defend their order: the day (Shabbat) causes the wine and the meal, and besides, Shabbat is already here, and the wine is only now brought to the table. However, Beit Hillel argue that the wine causes the sanctification, since if there is no wine, the Kiddush is not said. Moreover, wine is more common than Shabbat, and the rule is that things that are more common take precedence.
When to wash one's hands? Beit Shammai say that one should first wash the hands, then pour the cup of wine and say the Kiddush, and Beit Hillel reverse the order. Here the disagreement is about the intricate laws of ritual purity. Beit Shammai say that if the hands are not washed, the liquid that is on the outside of the cup may accept impurity from the hands and then transmit it to the inside of the cup. Beit Hillel argue that if you do wash the hands first, then the liquid on the outside of the cup may transmit impurity to the hands, and nothing is gained by washing them! However, according to Beit Shammai one may not use a cup whose outside is impure.
The other disagreements are where to put the napkin, when to sweep the floor, and how to properly say the Havdallah - the separation between Shabbat and weekday, after Shabbat has ended, and for all of them the Talmud finds the underlying reasons.
Art: Pierre Auguste Renoir - Sweeper