I would have to disagree slightly with the previous submissions, just based on my own personal experience. These idiomatic prepositional words can be very tricky when learning a new language but I will do my best to explain their general meanings.
in hand (under control, in procession of): A bird in the hand is worth two in the bush.
at hand (something will arrive or occur but has not occurred yet) - also not used very much from my experience: the time is at hand.
on hand (something is available, accessible, near) this one I have experienced before and used myself: do you have any tissue on hand by any chance?
I believe some people may use at hand and on hand in similar situations to express similar meanings to the two can be interchanged depending on one's dialect. But again this is just my thoughts.
here are some examples from real life English texts...
There were plenty of big names on hand
plenty of supporters were on hand to greet him
President Mahmoud Ahmadinejad was on hand at a Tehran nuclear facility
Give me a dozen of whatever you still have on hand
because doctors do not have the medical equipment at hand to save them.
inventing combinations that are appropriate for the situation and the people at hand
do you intend to know when the correct moment is at hand?
It is not as if other solutions are at hand
As soon as he's got the survivors in hand
He exits again with paper cup in hand
I responded, and walked to the docket, paperwork in hand
the cloaked figure approaching him, pistol in hand
(All examples from BYU)
Hope this helps,