General Synonyms:
* Dismay
* Disappointment
* Humiliation
* Mortification
* Embarrassment
* Frustration
* Vexation
* Irritation
* Displeasure
More specific synonyms:
* For a feeling of annoyance or displeasure: Irritated, vexed, annoyed, peeved
* For a feeling of humiliation or embarrassment: Mortified, abashed, ashamed
* For a feeling of disappointment or frustration: Discouraged, disheartened, crestfallen
The best synonym depends on the specific context. For example, if someone is feeling chagrin because they failed an exam, you might use disappointment or frustration. If someone is feeling chagrin because they made a public mistake, you might use humiliation or embarrassment.