How to define when a single apology or apologies are to be used in the case of a singular personal mistake? Is using my sincere apologies valid if there are multiple recipients or if you want to reiterate on a single apology. If the conversation is one to one should it be "my apology" for a single instance of a mistake.
"My apologies" is kind of a set phrase that can apply to a single offense toward a single person, or multiple offenses / multiple people. You can use it anytime.
While there is no rule against it, you would really only use "apology", singular, if you want to talk about a specific, singular offense. Example:
I want to offer an apology to you and your family for the way I acted at dinner last night.
It would be just as fine to say:
I want to offer my apologies to you and your family for the way I acted at dinner last night.
An apology is when you say sorry to someone, recognise your error.
Offering your apologies is one way to apologise. However, it can also refer to apologising on several occasions.
"I would like to offer my most sincere apologies for the inconvenience my error has caused."
That is a way to apologise, and could thus be called an apology.
That's a much more succinct way of using apologies in an apology.
That is incorrect. It's just not said, and sounds like you don't know what you're saying.
Whether you offer a person or people my (sincere) apologies or an apology is largely a matter of choice although this choice is influenced by the context.
Plural apologies does not imply numerous faults or errors although an apology does generally point to a single mistake, blunder or misbehaviour.
Both are expressions in which the speaker recognises that he/she has made an error or been guilty of some undesirable action and expresses regret to the injured party.
Both are more likely to be used in acknowledging social faults, errors of judgement and minor failures rather than of crimes.
Criminals who show real regret and sorrow at their actions are said to express remorse, a recognition of wrong-doing that is often taken into account in sentencing.
Grammarly.com points out that the expression my apology (as opposed to an apology or my apologies is likely to refer back to an apology that has already been made; it also illustrates which expression best fits which circumstances.
Dictionary.com also provides helpful examples of the most appropriate use of each.