A very polite, formal apology used in business or when expressing deep regret ('I'm truly sorry' / 'there is no excuse').
本日はご迷惑をおかけし、申し訳ございません。
I apologize for causing trouble today.
商品の不備があり、誠に申し訳ございませんでした。
We sincerely apologize for the defect in the product.
納期が遅れまして、大変申し訳ございません。
We are very sorry for the delay in delivery.
Sometimes used to express humble gratitude for someone's help (often combined with an apology), acknowledging trouble caused while thanking them.
お手伝いいただき、申し訳ございません。大変助かりました。
I'm sorry to have troubled you with this, and I appreciate your help — it was a great help.
ご多忙のところ対応してくださり、申し訳ございません。感謝します。
I'm sorry to have bothered you when you're so busy; thank you very much.