When you say catchment, are you outside a formally defined catchment, or do you mean that you live further away than the greatest distance offered?
Because the first thing you need to do is look at the rejection letters, which should give the reason why they could you not make you an offer. If there are catchments, and you have been wrongly classified as an out-of-catchment applicant when you are actually in-catchment, then that mistake could have meant you missed out on an offer that wouid have been yours if there had been no mistake. That wouid win an appeal.
Also check if the distance from school to home looks right (they don't get it wrong very often, but it happens).
You also need to check she is on the waiting list for any and every school you wouid find acceptable.
Bear in mind that the council only has to find you a place, it doesnt have to be a place you like. If over 3 miles (and there was no nearer alternative) then the council must provide transport, which might make the schools with vacancies acceptable. If you turn down a school offer, the council is under no obligation to find you another but you apstill need to ensure your DC is in education in September.
Back to appeals!
If there is no error, you are appealing on the balance of prejudice - ie that the prejudice (detriment) to your DC outweighs the detriment to the school, and all the other pupils by going over numbers. The strength of the case of the school will vary, you need to make your case as string as possible.
The key question is what is it about this school uniquely that makes it so important that your DC attends. Remember you are appealing for the wanted school, not against the one you've been offered.
Exam results generically won't count in an appeal, nor will transport (unless there are special needs that have been overlooked)
So it's the specifics - is your DC musical and this school has ensembles and orchestras and offered one doesn't? Or offers a language your DC has already started to learn? Or triple science? You need to be able to evidence your DC's skills/aptitude, and show why onkymthe sppealed-for school offers what they need