Well, first of all, do not, and I repeat not use the motivation that your assigned school is awful in the appeal:
a) this is not the way to make the LEA take kindly to your request
b) this absolutely does not explain why your child deserves the good schools any more than any other child in the area
Remember, it is only to you that your child is specially important: in the eyes of the appeal panel, every child is as deserving as any other child.
Imagine if you were on this panel: how would you react to a parent who wanted you to overrule your decisions "because it's more important that my child goes to a good school" than that other children do- would that make you favour that child above the 300 others whose fates you also had to decide?
Besides, having many children with SN/from a disadvanaged social background does not make a bad school and you will absolutely not make friends anywhere (certainly no on Mumsnet!) by saying so. Some of the best state schools in the country fit this description. There is no way any self-respecting LEA will take this seriously.
Instead, you have to focus on why your particular child needs to get into this school. First of all, check the admissions criteria. The most common way of getting in on appeal is if you can prove that the LEA have failed to follow their own criteria, in other words let another child in when yours should have been higher on the list according to their own stated criteria. A common order for admissions is:
Children in care.
Statemented children.
Children in catchment with medical or social needs.
Children in catchment with siblings already at the school.
Children in catchment ranked according to distance from school (and every LEA has its own way of measuring which you have to check out).
Children outside of catchment with medical or social reasons.
Children outside of catchment with siblings already at school.
Children outside of catchment ranked according to distance.
But remember that the LEA can apply the criteria correctly and still not be able to admit every child who is in one of their top categories, as the schoo quite simply gets full. Also bear in mind that if the LEA consistently start over-filling schools, the good schools will not remain good schools. And that in Infants, there are rules about class size.
Even if the LEA have applied their criteria correctly, you can still have a chance of winning an appeal, if there are very special reasons (usually medical or social) why your dd needs this particular school. We got dd into (admittedly) secondary school by showing that it was the only school that could meet her medical needs and we're hoping to do the same for ds; a friend argued that their preferred school was the only one to offer special counselling support that her dd needed. Anything like this has to be documented and supported by expert evidence.
Oh but I suppose children like ours are exactly what would put you off a school.