Hello OP. Happy New Year to you too!
Having three to six months' worth of expenses saved is a good rule of thumb. However, each person's circumstance is different.
Emergency fund is a must. You could argue how big should it be, but if it's also for the purpose of house repairs and job loss, than 6-months worth, seems like minimum. Although it might be a better idea to figure out a depreciation rate for all your assets and transfer the equivalent of it (+ a bit more on the top) to your emergency fund on each pay-day. Such fund then could grow to more than six-months worth, but could be used for major works like house renovation etc.
Of course, holding it as cash or in no-interest bank account would be a waste, given the inflation rates are still quite high, so you could combine that with your point 2 - investing and invest it somewhere, provided it's a type of investment that allows you to divest in a relatively short time. Otherwise it would be useless as an emergency fund.
One of the largest and most common mistakes people made in the previous year was not investing.
Everyone who has any savings or any surplus wealth is invested in something. Often time they're invested in cash money, one of the worst types of investment, that is designed to lose its value through inflation.