I want to do laptop maintenance and looking to ground myself with an anti-static wrist strap.
How, or where, do I ground myself to the laptop? Where can I find a grounding point in order to not make a mistake and spoil something?
You do not want to ground yourself to the laptop. You want to be grounded to earth.
The anti-static wrist strap should have come with an attached wire. The other end of that wire should be connected to an electrical ground. The center screw of the AC wall outlet holding the cover plate is a suitable connection point that is tied to your dwelling's ground (assuming you have 3-prong outlets and the outlet is properly wired and up to code). Make sure you attach to bare metal and not the paint on the screw.
Once you are grounded, you do not want to touch any live circuit. So the equipment should be turned off and unplugged. Ideally, especially with high voltage devices, you should use the hand that has the strap to poke around and the other hand behind your back or in your pocket. The idea is that if you do get shocked, the electrical current only has a path from your hand to the strap, and not across your chest and heart.
The purpose of grounding is to allow electrons on you to flow to an object so that the voltage difference between two objects is 0. This is so you don't dissipate the voltage directly onto the motherboard and other vital components.
Mike Insch does not understand the concept of ground. A ground is merely a reference point for which other objects voltages are compared to. The electrical earth might actually have a different electrical potential from that of the chassis. This could very well be the case in laptops with only two prong power supplies
A laptop chassis can be a ground, as long as it is on a good conductor. Those plastic chassis won't work. Also, it is really difficult to find an appropriate place to clamp the wrist strap. Also, I have had many laptops power supplies only have TWO PRONGS. There is no connection to the electrical earth.
The laptop should not be worked on while plugged in or with the battery in not because of lack of earthing, but rather because there is current still flowing through the motherboard.
Theoretically, the entire idea of the strap is so that you & your work are at the same potential. This isn't to protect you from mains voltage [there is some protection in one of these straps, but common sense of disconnecting from live mains is preferred], but to protect the circuitry from any damaging static charge you're carrying. Mains Earth [US: Ground] is as good a leveller as any for this purpose, if everything is connected to the same point.
Your actual issue for a laptop is that the power supply doesn't actually connect the laptop to earth at all, even if the external 'transformer' section has an earth pin to a good mains earth. It may have sufficient leakage from signal ground to mains earth to keep it potentially level, but it's a side-effect rather than an intentional provision.
In theory, this makes it 'not a good solution'. In practise, however, there's a fairly broad tolerance.
To get yourself down to ground potential you can just grab a tap [faucet]. If you don't get a tiny shock you weren't far off equal potential anyway. So long as you don't run around a nylon carpet wearing plastic shoes & a nylon shirt, rubbing a cat on your front on a really dry day, you're unlikely to build up enough charge to cause any issues between doing that & getting back to your laptop.
At that point, you can clip your wrist strap to any convenient metal part of the computer. Leaving the charger connected to the mains but with the switch off might help towards keeping you all at the same potential, even though there's no real path to earth between laptop & wall plug, but you're not going to really get too far from that water pipe ground zero anyway. If you have a desk with metal framing, even that would be enough to connect you & your work to, in absence of a constant real earth.
You'll be fine.