at/on (the) sight of sth

According to Collins COBUILD Dictionary CD-Room 2006:
The sight of something is the act of seeing it or an occasion on which you see it.
I faint at the sight of blood…
The sight of him entering a room could flood her with desire.
If you catch sight of someone, you suddenly see them, often briefly.
Then he caught sight of her small black velvet hat in the crowd…
= see

If you say that something seems to have certain characteristics at first sight, you mean that it appears to have the features you describe when you first see it but later it is found to be different.
It promised to be a more difficult undertaking than might appear at first sight…
If something is in sight or within sight, you can see it. If it is out of sight, you cannot see it.
The Atlantic coast is within sight of the hotel…
My companion suggested that we park out of sight of passing traffic to avoid attracting attention.
– If you lose sight of an important aspect of something, you no longer pay attention to it because you are worrying about less important things.
In some cases, US industry has lost sight of customer needs in designing products…
= forget

If someone is ordered to do something on sight, they have to do it without delay, as soon as a person or thing is seen.
Troops shot anyone suspicious on sight…