They all mean the same thing but only the first one sounds right. "In case" means there is a chance it will rain but that is not certain. "Should" standing alone means a rain is expected. "Might" standing alone means a chance again, maybe 50% more or less. So when you put "in case" and "should" in the same sentence you have contradicted yourself with two different possibilities. When you put "in case" and "might" in the same sentence you are saying the same thing with two different modifiers. Take your umbrella in case it rains. (a chance of rain, advising you to take an umbrella) Take your umbrella, it should rain. (rain is expected, advising you to take an umbrella) Take your umbrella, it might rain. (a chance of rain, advising you to take an umbrella)

'it rains' is the present tense, like 'When it rains, I use an umbrella'. 'it should rain' is when a person already knows it will rain, but there's still a small chance it won't. 'it will rain' means it definitely will rain. In this case, all of your sentences mean almost the same thing because of the 'in case' part. The first one sounds normal, the second more formal, the third a little awkward but more like the first. Sorry if this is confusing! I'm still getting used to explaining things.

“it should" or "it might" both sound bookish. I think that's the "subjunctive mood"? Maybe they still say it in UK, but I think in modern English (Both UK and USA) you will never hear this. 就說，"it might" 跟“it should" 好像過時的說法。只有語法書會友這些說法。

Take your umbrella in case it rains= when its raining Take your umbrella in case it should rain= 100% it will rain Take your umbrella in case it might rain= 50% it will rain i think this!!