I got usage of すみません versus ごめんなさい from here: Usage of すみません (sumimasen) versus ごめんなさい (gomen'nasai).
I heard Japanese people use 申し上げない (or 申し上げません) while talking on the phone. I would like to know the proper usage of 申し上げない and 申し訳ない in contrast with すみません or ごめんなさい.
There is no such thing as
申し上げない
申し上げません
for apology. You probably wouldn't ever say that anyway even though it's grammatically correct being the negative form of 申し上げる. It is not the meaning you think it is.
What you were asking is probably about
申し訳ありません
It's a formal apology that you would use after making a mistake in a formal situation i.e. a workplace.
You could use
申し訳ない
in informal situations as well. For example, if you caused your friend some trouble somehow, you could tell them 「申し訳ない」.
The distinction between
ごめんなさい
and
申し訳ない
becomes apparent here:
ごめんなさい is for when you want to apologize for your actions.
すみません is when you express your feelings regarding your mistake.
申し訳ありません is when you want to express your sincerity and regret at your mistake; you're literally at a loss for words regarding your mistake if you say this.
The focus of all these is different. If you done goofed towards your friend or caused them lots of inconvenience, it would be appropriate to say 申し訳ない to them. Similarly you could say 「すまん」.
(This is a good example: http://okwave.jp/qa/q6530648.html )
(By the way, someone else in their answer used the kanji for すみません, but that is an unusual and uncommon thing to do and I highly recommend against it as it looks very out-of-place.)
customer: I have received the parcel from you today, but it's color is red. I orderd bule one. seller: 申し訳ありません。青い商品をすぐに発送いたします。I'm sorry. I'll ship the blue one immediately.
The direct transration of "申し訳ありません" is "There is no word to express my apology." So it is used to admit the fault completely. Then a compensation is sometimes expected.
"申し訳ない" is casual expression. You can't use it in the above situaion of the example context.
When you step on someone's foot in the train. You should immediately say "済みません"
The direct translation of "済みません" is "There is no way to appologize" but "済みません" can be often heard everywhere in Japan. It's casual phrase to express your excuse for not serious mistakes.
