だが details

Examples

but; and yet — sets the following sentence against the one just closed

何度なんども彼かれに電話でんわをかけた。だが、一度いちども出でなかった。

nando mo kare ni denwa o kaketa. daga, ichido mo denakatta.

I called him again and again. But he never once picked up. neutral

計画けいかくそのものは悪わるくない。だが、今いまの予算よさんでは実現じつげんできない。

keikaku sonomono wa warukunai. daga, ima no yosan de wa jitsugen dekinai.

The plan itself isn't bad. Yet with the budget as it stands it can't be carried out. neutral

ご要望ようぼうは確たしかに承うけたまわりました。ですが、規定きていにより今回こんかいは対応たいおういたしかねます。

go youbou wa tashika ni uketamawarimashita. desuga, kitei ni yori konkai wa taiou itashikanemasu.

We have certainly noted your request. However, under the rules we are unable to accommodate it this time. formal

しかし
しかし carries no style of its own and sits equally well in ですます prose, a speech or a plain-style essay; だが has the plain copula だ inside it, so dropping it into a ですます passage breaks the style — which is why editorials and fiction take だが and a polite report takes しかし or ですが.
ところが
ところが needs the following sentence to defeat an expectation the reader is already holding, so it reads as a twist; だが marks any opposition at all, including a flat, unsurprising one (賛成さんせいも多おおい。だが、反対はんたいも根強ねづよい), where ところが would promise a surprise that never arrives.
だけど
The けど／が split survives the move to the head of the sentence, so だが is plain-style WRITING (editorials, narration, commentary) while だけど lives in speech and messaging and turns up in print mainly where the writer is imitating a voice.

が
The same が, one sentence boundary apart — inside a sentence が welds onto the end of the first clause (検討けんとうしたが、見送みおくった), while だが is that が with the copula frozen onto its front, standing alone before a clause whose previous half is already closed with a full stop.