Your formatting leaves much to be desired. However, you can simplify things by checking the values as you go through the initial loop, like this:
flavours = ["Vanilla", "Chocolate", "Mint Choc Chip", "Rosewater", "Strawberry", "Mango", "Frutti Tutti", "Fudge Brownie", "Bubblegum", "Stracciatella"]
Ice_Cream= [0]*10
print("Please input your score on these flavours of ice cream. With 1 being the lowest and 10 the highest.\n")
high = 0
index = 0
for i in range(10):
print(flavours[i]+":")
Ice_Cream[i] = int(input())
if Ice_Cream[i] > high:
high = Ice_Cream[i]
index = i
print("\nResults:\n")
for i in range(10):
print(flavours[i]+":",Ice_Cream[i])
print ("\nYour favourite flavour of ice cream is", flavours[index])