いいえis the super formal way to say “no”, but it is very rarely used colloquially.

Most commonly, people sayいやor even more commonly shorten it toやto say “no’.

In more formal situations (e.g. a job interview), you can sayいえwhich is shortened also shortened into a single mora.

やだmeans “don’t want to”, but this has a childish connotation associated with it.

When a child throws a tantrum, it’s called 駄々をこねる1. This means to not accept some reality due to selfish reasons.

Footnotes

  1. 「捏ねる」literally means to twiddle with your thumbs.