I'm sorry; I apologize
A formal, polite apology used in business and serious situations. Stronger than すみません.
I am sorry for causing you trouble.
Used to express deep gratitude, often when someone has gone out of their way to help. Conveys a sense of indebtedness.
Thank you very much for coming all this way.
I apologize for the late reply.
ごめんなさい is a personal, casual apology used with friends and family, whereas 申し訳ありません is the formal expression expected toward superiors, customers, or in business settings.
申し訳ない is the plain, less formal adjective form, while 申し訳ありません is its polite conjugated form used in formal speech.
申し訳ありません is a strong, direct apology that admits fault, while 悪しからず simply asks the listener not to be offended by unwelcome news.
申し訳ありません is a more formal, weighty apology suited to business settings, whereas ごめんください is rarely used this way in modern speech.
申し訳ございません uses the more formal ございません instead of ありません, making it an even higher-register version of 申し訳ありません for very formal or customer-facing situations.
申し訳ありません is the very formal, humble counterpart of ごめんなさい, used in business or serious situations rather than everyday ones.
Standard polite apology; the most common written form.
Alternate spelling using kana for わけ; less common but still seen.
From 申し訳 (もうしわけ, 'excuse') + ありません (polite negative of ある, 'to exist'), literally 'there is no excuse'.