Well for me personally, we would hardly use those in casual conversation except for loyalty. 'Allegiance' is commonly used more in literary works, but we don't use the word fidelity. We use it's opposite more: infidelity. Also, all of them are synonyms for loyalty. Sentences: 1.) She showed immense loyalty to her friend. 2.) Her allegiance to her friend through their childhood made them best friends in adulthood. 3,) His infidelity caused their marriage to end. (*Note I used infidelity) So in short: 'Allegiance' is used more in writings and such (like essays, books, etc.) Fidelity is not used at all. It's opposite: infidelity is used. Loyalty is the most commonly used out of the three. This is used in conversation and books! I hope I answered your question! If you have anymore just ask~

I will write this answer in part responding to Carita's answer. "Fidelity" is in fact used. Its opposite, the word "infidelity" is used almost EXCLUSIVELY to describe marital/sexual infidelity between people in a relationship. So, "fidelity" is often used in reference to human romantic/sexual relationships. The one other case you see it used is as a TECHNICAL term in film/computer graphics to describe "picture fidelity", meaning how "true" a picture or photograph is to the actual real-world object(s) or scene pictured. Any usages of 'fidelity" outside of these two are likely more old-fashioned. Allegiance has to do with alliances. It has more to do with official signed agreements between states and political heads than it does human emotion. It does not necessarily match what you feel in your "heart". You can hold an allegiance reluctantly, or pledge allegiance half-heartedly or while intending to turn your coat later. Loyalty is the purest/most plain form. It is essentially what a dog feels for a good owner or what best friends have for each other.