Sorry...
The most common way to apologise as it can be used in many situations, for example:
-When we accidentally bump into someone
-When we know we have done something wrong ('Sorry I'm late!')
How careless of me!
Used when we criticise ourselves for making a mistake, for example:
'I took your notebook by mistake! How careless of me!'
I shouldn't have...
Used when we realise we have done something we regret, and we wish we could go back in time, for example:
'I shouldn't have talked to the teacher like that.'
It's my fault.
We may say this when we want to take responsibility for something, for example:
'It's my fault that I didn't stop my dog from eating my homework.'
Please don't be mad at me.
A rather informal phrase to ask for someone's forgiveness, for example:
'I'm sorry we have to cancel our lunch date. Please don't be mad at me.'
I (would like to) apologise for...
A formal way of saying sorry. This is commonly used in formal/ business emails, for example:
'I would like to apologise for not fulfilling my responsibility.'
Pardon me/ excuse me
Used when you want to interrupt a conversation or ask for somebody's attention, for example:
'Pardon me, but can I just butt in here?'