Within Nethack 3.3.1, there are several varieties of cockatrice (all show up as a 'c'). There is the 'chickatrice', the classic 'cockatrice' and the pyrolisk. All of them are visible in infra-vision and naturally hostile. The chickatrice is a 'tiny' monster, while the cockatrice and pyrolisk are 'small' monsters.

There is also a special type of 'zoo' that can be created called a Cockatrice Nest (other special zoos are courts, barracks, morgues, beehives, leprechaun halls, and ant holes). These will only be created at depth greater than 16 and have a 1 in 8 chance of being created. Upon entering you will get the message "You enter a disgusting nest!". There is no special level message.

From this point on, when revering to the 'cockatrice' I am referring to the cockatrice and chickatrice only - not the pyrolisk.

The cockatrice has attacks that may stone you - and not in the psychedelic way (use violet fungus for that or exploding black lights for that). If you hear a hiss, there is a 10% chance that the stoning will begin (and not in the martyr way). However, if it is a new moon, and for some reason you are not carrying a lizard corpse, then the stoning chance is 100%.

So, after a brutal combat with long range weapons, you've killed a cockatrice and there is a corpse. What can you do with it?

Eat it
Eating a cockatrice corpse is instant death. Most things related to cockatrices are instant death. Do not eat the eggs of a cockatrice - that is certain death.
Pick it up/Kick it
Touching a cockatrice corpse with a 'bare' part of the body will kill you. If you aren't wearing gloves and touch one, you die. If you aren't wearing boots and you kick one, you die. If you are blind and not wearing gloves and fumble around in the dark and find one, you die. I hope you can see a trend here.
Wield it as a weapon
However, if you are wearing gloves, you can pick up a cockatrice corpse and it is now part of your inventory as a '%' (food). It is still not recommended to eat it. You can wield it as a weapon.

<Insert picture of mighty barbarian running around with a dead chicken in his hands here>

What happens when you hit a monster with a cockatrice corpse? Same thing that happens to you when you touch one yourself - they get turned to stone. This often results in a statue of the monster sitting around, but if the monster is a golem it will become a stone golem. Yes, the DevTeam thinks of everything.

This technique also works with cockatrice eggs. However, remember that you penalize luck if you were the mother. These have been nicknamed 'stoning grenades'.

Realize the greatest danger of walking around with cockatrice corpses is fumbling. Fall in a pit? Set off a landmine? Find a pair of cursed gloves cheap? Fall down the stairs? Well, you're a statue now too - and there's no time to eat a lizard corpse.

The statues that are created from stoning a monster (or player) will only contain the items in that being's inventory and never a spell book (unless one was carried).