Whether you know it or not, you probably can get away with not having an account at a normal brick-and-mortar bank like Wells Fargo, Chase, or Bank of America. Cut the cord with them just like you did with cable TV, and switch to either an online bank or a credit union.
Brick and Mortar Banks Suck
Credit unions offer better customer service. At Bank of America, you're just a number. A credit union serves the local community, and will give you better rates on mortgages and car loans.
Money orders are almost universally as acceptable as cashier's checks. If you're giving the title company a down payment on a house, you'll know it in advance. If you're paying the security deposit on a new lease, you can get money orders at Wal-mart for less than a buck.
Notarizing documents is a service available elsewhere, like at the UPS Store. The most common need for a notary public is for legal documents, anyway, and many attorney's offices have one on staff for that reason.
Online Banks Raise the Bar
Online banks offer better ATM access. Schwab Investor Checking (available free with the opening of a second brokerage account which you don't have to fund) will refund all fees you are charged at ATMs anywhere in the world. That's right, the ATM charges you a $7 fee and within a few days Schwab gives you a refund for that $7. On top of that, Schwab also charges no foreign transaction fees, which is a lifesaver when traveling abroad.
Online banks offer check deposits on mobile devices. However, there is always a maximum amount per check or per day they'll accept. The limit varies, but it's normally in the four-figure range so if you get checks you need to deposit of $5000 or more (that you can't arrange direct deposit for), you might be one of the few people that needs a brick and mortar bank.
Online banks are more innovative. Capital One 360 used to be known as ING Direct. Fortunately, when they were bought out, Capital One kept all the innovation they had pioneered, like the ability to have an unlimited number of savings accounts, and getting a small pack of checks (if you need them) for only $5.
Online banks are actually free. Yes, if you meet the requirements by having direct deposit and making a certain number of debit transactions every month, you can avoid the monthly fees at a brick and mortar bank. But those requirements are there because banks can't afford to offer everyone free accounts, no questions asked, like they did a decade ago (largely because legislation has curtailed their ability to extract profit from free customers through exorbitant fees). They have to pay for the overhead of physical locations and all the staff and anciliary costs that come with it. Online banks don't. In the case of online banking, free really means free.
First of all, an honorable mentions goes to USAA, because they deserve it. USAA is really more like a national credit union than a bank. They offer excellent services for their members, including great discounts on combining insurance products like home, car or renter's insurance. Unfortunately, they have a requirement that you or a family member is or was in the US Armed Forces, so they're unavailable to most of us. If you're eligible, though, they're definitely worth a look.
Finally, the winner. In this humble author's opinion, the innovative features offered by Capital One 360 slightly edge out a victory over Charles Schwab, especially if you don't mind looking for an AllPoint ATM (there is almost always one near you) and don't often travel outside the US (both benefits of Schwab Investor Checking).
Capital One 360 also has better support for transfers to external accounts and has an especially cool way of handling overdrafts. Overdrafting is off by default (so charges are simply declined if they would bring your balance below zero), but you can request to have it turned on at which point you are given a limit of $500 or $1000. You're required to bring your balance above zero by the end of the next month, and they only charge you interest for the number of days you were overdrafted. So if you go under by a couple hundred bucks, and it takes you a couple weeks to get it above zero, you are charged something like a dollar in interest. Way better than a $35 overdraft fee, that's for sure. Add on the low fee for getting checks, great support for transferring funds to your accounts at other banks, and unlimited savings accounts that you can create at the click of a button, and you'll wonder why you didn't switch to an online bank long ago.