With interest rates at online banks so much higher than their brick & mortar counterparts, many of us (myself included) choose to store our emergency savings at an online bank. There is of course one huge drawback- doesn't an emergency sometimes mean you need money right away? Most online banks don't give you access to checks or an ATM card so you might have to wait a few days to transfer money from your savings account to your checking account.
One solution to this is just to charge the emergency then immediately pay off your card from your savings account. I think this is the best solution for most people because it allows you to avoid the need to immediately transfer money. But what if you encounter an emergency that needs cash? One that I can think of is having your car towed- it costs around $100 to pick it up and they usually only take cash. If it's not in your checking its time to hope that you have a friend who owes you a favor.
Because of this I think it's important to spend some time thinking about what kind of emergency spending situations you might come across. I know you can't predict the future and some expenses come out of nowhere but I think there are some common financial emergencies that everyone needs to prepare for. If you figure out which ones apply to you, you will have a better idea of how much you need to save and how accessible that money needs to be. By predicting the unpredictable you can get out of the vicious cycle of using credit cards when you come up short on cash.

• Car Expenses: I feel like this is one of the hugest benefits to not having a car (I don't have one). For most people their #1 emergency expense is car problems. Have some kind of unpredictable problem with you car and all of a sudden you need to come up with $500 (or more!). If you drive a car you are pretty much guaranteed to have some kind of unexpected issue with it, whether it is something minor like a dead battery or something huge like getting your car smashed up in an accident. Either way, it is just smart to expect this and save up for it. Don't count yourself out just because you don't have a car though, if my bike was stolen I would want to buy another one asap.
  
• Medical Expenses: I can't even think of a scenario where you would need cash for medical expenses so this one is ok to go in an account that isn't immediately accessible. I recently spent around $125 for a common dental procedure despite the fact that I have dental insurance. Insurance only takes you so far.
• Home Repair: This one can of course be avoiding by renting! But if you own a home make sure that you have a cushion for repairs and maintenance (this is why it's so important that you get a mortgage payment that isn't draining you of every extra available cent). Ideally you'd want to have at least $1,000 saved for emergency home repairs, but even a few hundred is better than nothing. The more you can write a check for the less you will have to pay interest on if you use a loan or credit card for the cost.