Are savings accounts worth it?

Absolutely. There are two main reasons I suggest people keep money in savings accounts, regardless of the interest (or lack thereof). On a side note, not too long ago (within the last decade or so), you could find even plain savings accounts (particularly those at online only banks) that were paying interest in the 4-6% range. Obviously that is not the case now, and probably won't be for the foreseeable future, but we may reach a time in the next decade where savings accounts are paying more than 1% or less.

Moving on, the first situation where is definitely makes sense is for an emergency fund. You should have an emergency fund that could cover a minimum of 3-6 months of living expenses (and possibly more depending on your particular situation). That money should be kept in an account where it is easily accessible (i.e. liquid) and not subjected to the inherent uncertainties of the markets. To me, a savings account is generally going to be the best option because you can have it at the bank where you have your checking account or you can open it at another bank (perhaps an online only bank that pays more interest than a brick and mortar bank) and link it electronically to your checking account.

You could also keep an emergency fund in a Roth IRA since you can always remove your contributions without tax or penalty implications. However, if you go this route, I would still recommend keeping the portion intended to be your emergency fund in cash and not invest it.

The second situation is for money that you know for sure you will need within the next 3-5 years (e.g. down payment for a house, new car, etc.). Too much can happen in a short period of time in the markets to subject money that you know for sure you will need to the markets' uncertainties. Therefore, I usually suggest that money earmarked for certain use in less than 5 years be placed into something "safe" like a savings account.

In both of these instances, you are giving up returns for the knowledge that your principal will be fully intact if/when you need the money. In other words, if you invest your emergency fund or money you know for sure you will need within the next 5 years or so, when the time comes to make a withdrawal, there could be less money than you want/need (though there could be more if invested), which can be a bad situation in which to find yourself. So, you are trading security for return, and as long as you are doing that for the right reasons, it absolutely makes sense and is "worth it."

Yes, it is important to have money in savings for an emergency.  An emergency fund is liquid and available for us to us in case the unexpected happens. We are not looking to build wealth so we can sacrifice return to deal with an emergency if it arises.

Great question, one that I get on a regular basis.

There are two reasons I recommend keeping money in a savings account. The first reason is to seperate savings for particular goals or as a rainy day fund. The second is for safety and liquidity.

Most American households spend the entirety of their checking account on a monthly basis. Those who automatically or proactively move money from checking into savings are significantly less likely to overspend than those who don't. Considering savings as a high priority expense is a great exercise, and moving the savings into a seperate account puts a soft barrier you're much less likely to tap into for a new pair of shoes.

Even though there are alternatives such as CD's and bonds that will provide an additional rate of return, they are less accessible and may be subject to surrender charges (CD's) or price fluctuation (Bonds). Your emergency fund, or family safety net of 3-6 months of expenses should be in an FDIC insured account to which you have easy access.