This is a judgement call we all have to make individually. Each situation is different, and each individual will have a personal view. The important thing is to find a way which will cause you, and the chick, the least possible distress.
The bottom line is this question :
"Does this chick have, or is she ever going to have, a good quality of life?"
If the answer to that question is a definitive "No", then it's time to cull.
If you're still not sure, here are some more questions to ask yourself.
Ask yourself, too, whether you are holding off taking action for your own sake. It's hard to make this decision, but at the end of the day it's the kindest thing we can do for animals in our care if they are very unwell.
Do not, on any account, leave a chick in obvious incurable distress or pain because you can't bring yourself to euthanise her. That is unfair and unkind. It's heartbreaking, but the most humane thing we can do for a sick chick is to help her die in a dignified and peaceful manner.
When I had to cull a very sick chick, I carefully researched different methods before choosing the one which felt kindest and which I had confidence in my own ability to do quickly, causing the least possible distress. Before I discuss that method, here are some others I found which I could not, for my own reasons, use.
This is the quickest and most humane way to kill a very young chick. At this age (under two days post-hatch) the spine is soft and breaks easily.
How to do it: Using a pair of sharp scissors, simply snip off the head.
This is also sometimes called 'thumbing'. When done properly, it is another quick and painless way to cull.
 How to do it: flick the neck backwards over your thumb. 

How to do it: place the chick into an airtight container. Drop starter fluid onto an absorbent pad next to it.
How to do it: Take your sick chick to a veterinary surgeon who will administer a lethal injection.
Again, please note: this method is NOT suitable to cull anything other than an ill chick who is less than two days old.
This is the method I first chose when I had no choice but to cull. It's quick, easy to do and doesn't require skills I don't have.
It will not be to everyone's liking, and there are no studies on very young chicks to assess exactly how much pain it may cause.
Anecdotal evidence is that the chick is unconscious very quickly and dies in under a minute(1).
One of the benefits of this method is that you probably have everything you need in your store cupboard.
Arrange the smaller container inside the larger. I added a cloth bed for the chick to lie on, and a pad of paper towel underneath the smaller box, in case of spillage.
Add bedding and a paper towel.
Add the baking soda to the smaller box
Make sure the box closes tightly.
Writing a tribute to a lost chick (or chicken) can help with the grieving process.
Clicking on this image will take you to a page where you can leave your own tribute.
Tributes will be published as soon as possible, and if you so choose, you'll be informed when yours is live on this website.
Thank you for your patience.
Most chicks who hatch will be healthy and happy. Here are some articles about how you can make sure they stay that way.
J. K. Blackshaw et al : 'The behaviour of chickens, mice and rats during euthanasia with chloroform, carbon dioxide and ether'. Pub. Journal of Laboratory Animals, 1988.