
NEVER write it down!
I was at an IT security conference 25+ years ago and there was a "reformed" bad guy giving a talk on
"Physical Security" - a genuine, but often overlooked threat! He said when he broke into a home, place of work, (or hotel room), he would take about 30 seconds to sit down at his victim's desk and take a quick look around in drawers, on shelves, under keyboards - at anything within arm's length for a notebook, steno pad, index card box or just a piece of paper (even sticky notes stuck to monitors) for passwords. And very often he would find such a list. Then, instead of just grabbing and hawking the hardware with a "fence" or shady pawn shop, he would take that list too and use the passwords to see if he could access their bank accounts or obtain other personal information he could then sell/trade.
Years later when I started my own shop/consulting business, I would make occasional house calls. I started to do the same thing - sit down at their desk and take a quick look around and sure enough, I would often find them. Shockingly (to me anyway) I would discover some of my clients were using a password safe/manager only to write the master password down on a sticky note stuck to the monitor, or under the keyboard.
NEVER assume an unauthorized person will never "physically" sit at your desk, or rummage through your home. It doesn't have to be a burglar either. It could be someone visiting your home like a whiz kid mischievous nephew or one of your kid's nosey friends. Or even your own kid!

For a work computer, never assume your co-workers would never stab you in the back.
The biggest advantage to using a password safe is you only have to remember one [hopefully very strong] password - the one to unlock the safe. Okay two passwords - you also need to remember the one to unlock/wake/log into your computer - which everyone uses, right? RIGHT?
I do NOT recommend using the password managers integrated with most browsers. Again, it is about "physical" security. If they are stored in your browser and someone gains physical access to your computer, they may have access to all your important sites and accounts too.
I’ve been using
SplashID since my Palm Pilot days. Sadly, it is no longer free - otherwise I would heartedly recommend it. Other recommended safes include the very basic, simple to use, but robust and effective
Password Safe.
KeePass Password Safe.
Enpass and
RoboForm are very popular favorites too.
I no longer recommend the very popular
Last Pass. Due to its popularity, it has been the target of state-sponsored Chinese (and other)
hackers multiple times. While supposedly, they have not gained access to users passwords, they have been successful hacking the site. Not worth the risk to me.
This points out another advantage to the simple and effective "Password Safe". It is totally stand-alone. It does not rely on, nor does it store anything in "the cloud".