If it’s zero or one object, use singular. If it’s more than one, or an undetermined number, use plural.

In your example, “de pommes” should be plural because “de” is short for “des”, and “des pommes” is plural. Similarly, “aucune pomme” should be singular because “aucune” is clearly zero.

You need the plural in the first case je n’ai pas mangé de pommes ce matin because it negates a sentence where more than one apple is eaten (j’ai mangé des pommes.)

If you only eat at most one apple in the morning, you would say je n’ai pas mangé de pomme. This sentence also negates j’ai mangé de la pomme.

In the second one, it is singular je n’ai mangé aucune pomme. Note however that the plural is possible in the cases where the noun has no singular form, or if its singular form has a different meaning, e.g.:

Je n’ai pris aucunes vacances.