‘I’m sorry for what I did, but I’ve had a hard day.’  Nuh. That’s not taking responsibility. Avoid the word ‘but’ and don’t give excuses.  Try instead:  ‘I’m sorry for what I did.’   Good. Then think of ways to prevent a hard day from having such an influence over you in the future.

‘I’m sorry for what I did. I didn’t mean to upset you.’  Wrong. You’re implying the recipient is thin-skinned, and that you really shouldn’t be apologising for something so minor.  ‘I’m sorry for what I did. I regret what I did and apologise for it.’  Good. That’s more honest.

‘I’m sorry for what I did. I’ll try to ensure it doesn’t happen again.’  Wrong. Do more than try.  ‘I’m sorry for what I did. I will ensure it doesn’t happen again.’  Good. And, ensure it doesn’t.

‘I’m sorry. I didn’t realise it meant so much to you.’ That’s not an apology, that’s a sneer.
‘I’m sorry. That was important to you and I was grossly inconsiderate. I’ll be more considerate in the future.’  That’s better.

‘I’m sorry, but all I said was . . .’   No buts! And don’t undermine your apology by trivialising the incident. If you don’t think your behaviour is worth an apology, don’t be disingenuous by giving one.

‘We are closed for renovations. We apologise for the inconvenience.’  Wrong. Keep your apologies for when you’ve done something wrong. An honest and non-patronising statement might be, ‘We are closed for renovations’.

‘I’m sorry for what I did. I thought you had a sense of humour.’  That’s not an apology, that’s an insult. And, you’re blaming the person for not seeing the incident in the same way you saw it.

‘I’m sorry. I thought you might find it funny.’  Wrong. You are blaming the other person for not finding it funny. Try: ‘I was wrong and I’m sorry. I can see you were hurt and I regret it.’

‘I’m sorry if you’re offended,’ and ‘I’m sorry you took it the wrong way’ and ‘I’m sorry you feel that way.’  No. In each sentence you are not apologising for your actions, you’re simply regretting the way they interpreted it. You’re suggesting it’s their fault they’re offended.

‘I owe you an apology.’  Wrong, unless you immediately follow with the apology itself. Telling someone you owe them $20 is not the same as giving them the $20.  Try: ‘I apologise for —’

‘I wish to apologise for —’   Wrong. Wishing you were giving someone $20 is not the same as giving them $20.   Try: ‘I apologise for —’

‘I’m sorry, alright!!’ Obviously, no. That’s not an apology; that’s a demand to back off.

Here is a dissection of a real, but poor, apology.
I present it not to criticise the giver of the apology. He remains unnamed and I hope unrecognised. I don’t even know if the apology he gave was warranted because I don’t know the facts of the matter. You and I are examining this apology only because it’s a good example of a poor one.
For all I know, his apology might be perfect if we take into account the legal ramifications. I’m not criticising the man for not giving a better apology, I’m using his apology as an example of what not to do. In the legal world it can be a different matter, but that’s not our concern here.

‘I love the XX club. It’s been a part of my life since I was born. It was part of my father’s life, it was part of my grandfather’s life.
  And I am so sorry for anything that’s happened or that’s been done wrong to our players or been done wrong to our football club.
  I and we would never do anything intentionally to harm this club or to harm the game . . . that has given me so much and given so many people so much.’

Let’s examine his words.
‘I love the club. It’s been a part of my life since I was born. It was part of my father’s life, it was part of my grandfather’s life.’ 
  He is saying, ‘Please understand that I meant no wrong. I hope you will go easy on me.’ That’s acceptable.

‘And I am so sorry for —’
He apologises, but to whom? To no one. It’s as though he feels obligated to fling out an apology, and does so, without directing it to the people who might actually deserve it. We can feed someone either by placing the food in front of them, or by flinging the food onto the floor and letting someone find it. Which is the more sincere, the more respectful? Try instead:   ‘I apologise to A, B and C.’  That’s respectful, and it’s taking responsibility.
He could add something like: ‘I realise this affects everyone in the club, including the supporters, and I apologise to them.’

Moving on:
‘I am so sorry for — (what has) been done wrong to our players —’
  He appears to be apologising for what other people have done wrong. No one can apologise for what someone else has done without being disingenuous and patronising. Try instead (depending on the truth of the matter):
‘I will not apologise for the actions of others. However, I deeply regret that I did not adequately oversee . . . and I apologise for not doing so.’
Or, ‘I knew what other people were doing and I knew it was wrong. I apologise for condoning their actions —’
Or, ‘I’m sorry for what I did wrong to our players.’
Or, ‘I cannot apologise because I have done nothing wrong.’

Now let’s look at his whole sentence: ‘And I am so sorry for anything that’s happened or that’s been done wrong to our players or been done wrong to our club.’
‘anything that’s happened —’ Could he get any vaguer?
‘anything that’s been done wrong —’ He uses the word ‘anything’ rather than ‘everything’, putting in doubt that there even has been a wrong. It’s as though he doesn’t know what has been done wrong, but is apologising for it anyway. He’s like a child admitting to wrongs he doesn’t understand, just to get the fuss over and done with and to keep people happy. He appears hapless, not contrite.
If his apology were sound he might say something like, ‘I believe I have done nothing wrong and I apologise for nothing. However, I am distressed with how much suffering there has been in the club and I recognise that many people believe I am at least partly responsible for it. I will be doing everything I can to ensure — ’
  Or,  ‘I am sorry to A, B and C for —’ (citing specifically what he did wrong, one item after the other.)

Moving on:
‘I and we would never do anything intentionally to harm this club or to harm the game . . . that has given me so much and given so many people so much.’
He would not seriously believe that we would think he was intentionally harming the club or the game, so why would he say that? He says it because he wants to make himself look innocent, and to avoid taking responsibility. He could say something like (depending on the truth of the matter):
  ‘I and we have harmed this club and the game. On behalf of myself and (the others) I apologise for doing so.’ 
  Or, ‘With our negligence, I and we have harmed this club and the game. On behalf of myself and —’
  Or, ‘I and we would never do anything intentionally to harm this club or to harm the game, which is why I insist that we are innocent— .’

When I saw him give the apology I got the feeling he was weary of the accusations, and was throwing apologies at us to get us off his back. His distress was sincere, but his apologies didn’t seem to be.
   Perhaps he was genuinely remorseful. If so, he could work on his ability to give apologies.
   If he believes (perhaps rightly) that he did not owe an apology, it might explain why he gave such a poor one.

Would you like to know how to do it right? Try : ‘How to give a quality apology‘