Start with the places people actually keep wills, which are duller than you expect: a home filing box, a bedroom drawer, a safe, or with the attorney or solicitor who drafted it.
Work through the list below in order. Most wills are found in the first three places, and the later entries exist because occasionally they are not.
The search order
Work down it rather than across it. Each place is more effort than the one before, and most wills are found in the first two.
At home
The obvious places: filing box, desk, bedside drawer, safe, or a folder marked with anything official sounding.
The lawyer
Many firms store the original and issue the family a copy, so a copy at home may mean the original is elsewhere.
A safe deposit box
If they had one. Access after a death usually requires the death certificate and proof of your authority.
A will register
Where one exists. Some countries maintain a central record of where wills are lodged.
The executor
If a family member was named, they may already hold it and not have mentioned it.
Their adviser
Their accountant or financial adviser, who often knows whether a will exists even if they do not hold it.
Read the executor clause before anything else
When you find it, the first thing to look for is not who inherits. It is who is named as executor, because that person has the legal authority to act and everybody else does not.
If it is not you, several of the things you were about to do are not yours to do. That is usually a relief rather than a slight, and it saves a great deal of duplicated effort. There is more on what that role involves in being named executor.
If you find more than one
Generally the most recent valid will is the one that counts, and a later will usually revokes earlier ones explicitly. Do not destroy the earlier versions. They can matter if the newest is challenged or turns out to be invalid.
If two wills appear close together in date, or one is unsigned or unwitnessed, that is the point to get advice rather than to decide yourself.
If there genuinely is not one
Then the estate is distributed according to the intestacy rules where they lived, which are fixed and do not care what anybody intended. That often surprises families, because the rules rarely match what people assume, particularly for unmarried partners and stepchildren.
This is also the moment most people realise how much of the picture was never written down anywhere, which is a separate and larger problem covered in how to find someone's accounts.