We won an appeal to Colchester Royal Grammar this summer having polished our case very thoroughly having gone though the trauma of an unsuccesful appeal at another super-selective.
The school has old buildings and in years 7 and 8 a class size of 32. When we received the school's case it described a learning environment which seemed to contradict the glowing picture painted in the school prospectus. If this is the case with the school you are appealing to try and see if there are any inconsistencies.
We did not go in hard on the school. We said right at the beginning that we believed that the school tried to accommodate as many boys as it reasonably could but that the system was not perfect. We established how many appeals had been granted by other grammars using the same exam and admissions system and said that, as appeals were granted in those schools then clearly the system was not faultless. We managed to establish that, despite appeals being granted none of these other grammar schools ever seemed to go above PAN so granting an appeal shouldn't in fact prejudice the school. We went in hard appealing for justice for our son.
Clearly, corroborating what Etienne said above, the major part of the appeal was to demonstrate first that our son's acadmic ability was such that he was reasonable expected to achive a high enough mark to gain an automatic place; second, that his underperformance was exraordinary and unforeseeable; third, that we had strong mitigating circumstances to explain that underperformance.
We won, in part because we had a strong case but also, because the school's case was undermined. The PO, probably for no fault of his own, as he was presenting the Governors' case, came across as a bit shifty and the panel didn't seem to take to him. We didn't undermine on the big stuff but on little bits of detail. For what it's worth we now know that a lot of the things they said in their case are not true which is a bit concerning and had we known in advance possibly could have undermined school's case further. I am not sure that this would have helped us though. We tried to be reasonable and respectful and coureoug througout and let the panel draw their own conclusions. Some moderate of facial expression at the comments made were definintely picked up by the panel. They are not stupid and can draw their own conclusions and I do think it is vital that they connect with you and like you and sound of your DC. If you can get a chuckle out of them I'd say that goes a long way.
Don't know if this helps at all, if you want any more specific info feel free to PM me.