Number 2 (I will stay at home) shows us you’re referring specifically to YOU and what YOU will do. Number 1 could be you giving somebody else advice. I would stay at home (if I were you) Number 3 is more of a command, again to another person. And I’m sure you know but it’s STAY, not saty!

If + should + verb is a form of type 1 conditional. source: dictionary.cambridge.org/grammar/british-grammar/should [excerpt] We sometimes use should in hypothetical conditional clauses with if to express possibility. It is formal: [information leaflet in a hotel room] If you should wish to use the Internet, there is a code available at the reception desk. If you should decide not to go on the trip, you will get a full refund. [end excerpt] Given examples: 1 - If it should rain tomorrow, I would stay at home. This appears to be a nonstandard type 2 conditional. Standard version: If it rained tomorrow, I would stay home. 2 - If it should rain tomorrow, I will stay at home. This is standard and equivalent to the type 1 conditional "If it rains ...." 3 - If it should rain tomorrow, stay at home. This is standard and equivalent to the type 1 conditional "If it rains ... stay at home." "Stay at home" is a command to the listener [essentially, "If it rains ... you must stay at home."]

The three sentences you provided all use a conditional structure but have subtle differences in meaning: "If it should rain tomorrow, I would stay at home." This sentence uses the conditional "would", which indicates a hypothetical situation. It suggests that the speaker does not expect rain but is considering what they would do if it happens. Tone: Less certain, more speculative. "If it should rain tomorrow, I will stay at home." Here, the conditional "will" expresses more certainty about the future action (staying at home) if it rains. The use of "should" still adds a slight uncertainty about the rain itself. Tone: More certain about the action of staying home if the condition (rain) occurs. "If it should rain tomorrow, stay at home." This sentence is giving advice or an instruction to someone. The speaker is suggesting what the listener should do if it rains tomorrow. Tone: Directive or instructive. Summary: The first sentence is more hypothetical. The second sentence is more certain about what will happen. The third sentence is a suggestion or instruction to someone else.

I will comment on #1 and #2: "Will" is the modal verb that we use to express certainty. "I will stay" is a decisive statement about what you will do. "Would" is similar to will but far less certain. The meaning of "I would" is much less precise than that of "I will". It can mean too many things to list. "Would" can introduce uncertainty and conditionality not only about what will happen but to whom it will happen. Just to make matters more confusing, "would" can also express certainty, if the words are spoken with the right tone. For example, by placing a strong emphasis on "I", you could indicate that this is *definitely* something that you would do: "If it rains tomorrow, **I** would stay home."