thankfully; fortunately; by grace or providence
Polite set phrase used when reporting that things are going well or turned out well. It can sound like 'thankfully' or 'fortunately'; any religious sense such as 'by God's grace' is usually only background nuance in modern use.
Thankfully, my mother's condition has improved a lot.
thanks to your kindness; thanks to your support; thanks to everyone's help
Polite formula for acknowledging the listener's, customers', supporters', or others' help. The benefactor is often left unstated, so the phrase can thank the person being addressed or a broader group.
Fortunately, it looks like we will make it through the year safely again.
Thanks to your support, our shop has reached its first anniversary.
Thanks to everyone's help, we were able to reach our target amount.
おかげで is a plainer, less ceremonious way to say 'thanks to' a stated cause, while おかげさまで is the polite set phrase used to acknowledge good fortune or support.
幸い means 'fortunately' or 'luckily' on its own, whereas おかげさまで adds a polite sense of gratitude toward someone's help even when unstated.
ありがとうございます is a direct thank-you, while おかげさまで indirectly credits others' support while reporting that things are going well.
おかげさまで is the more polite, self-effacing way to say お陰で, often used to thank someone while downplaying one's own effort.
Commonly understandable kanji-mixed spelling using 陰 and 様; it can look slightly more formal or traditional than the kana spelling.
Attested kanji spelling using 蔭; less learner-friendly than the kana spelling but still a real written variant.
Mixed spelling with 陰 and kana さま; usable, though the all-kana form is usually safest.
Mixed spelling with kana おかげ and kanji 様; attested but less standard-looking than おかげさまで.
From お陰／お蔭, a polite form of 陰／蔭 meaning protective help, favor, or blessing, plus 様 and the particle で. Literally it evokes being under someone's or the gods' protective 'shadow,' but in modern Japanese it is a conventional polite phrase of gratitude or fortunate outcome.