I have no experience of primary school but we did appeal successfully for secondary (which is probably slightly easier as they are less restricted in extra numbers). Our experience was:
even so, very few appeals were successful (only a handful out of 45 to get into this particular school)
the admissions criteria were listed in order of priority, starting with children in care, then statemented children, then children in catchment with siblings at school, then children in catchment without siblings, then children out of catchment with special medical needs, then children out of catchment with siblings
the best way to get in was if you could prove that the LEA had not followed their own admissions criteria in refusing your child (e.g. you fitted category 2, but they had admitted children category 3 in preference to you)- but you also had to prove that you had submitted enough evidence in your original application to make it clear to the LEA that you should go in the category you thought
if it was proved that the LEA had failed to follow their own criteria, they had to admit you (not sure how this works for primary)
so first find out what those criteria are and go through them with a fine tooth comb
after this, you could still have your appeal admitted if you had not submitted this evidence, but the panel still judged that your child did have needs putting them in one of those categories which were so overwhelming that they outweighed the disadvantage caused to the school by being overfull and to the children already in the school (this will probably be much harder at primary level).
have to admit that though I knew my dd was a very deserving case (wheelchair bound and offered a place at a school with three floors and no lift!) but I still found this quite stressful
you have to remember that from the LEA's point of view, the fact that one school is better that another in academic or social terms is no reason why your child should have a place there in preference to all the other children in the authority
so you have to find other arguments
if you have a good argument, get evidence to back it up and appeal
if not, stay on the waiting list