[verb て-form] + もしょうがない / もしかたがない [i-adj 〜い → 〜くて] + もしょうがない / もしかたがない [na-adj] + でもしょうがない / でもしかたがない [noun] + でもしょうがない / でもしかたがない
てもしょうがない and てもしかたがない say that doing something is pointless: "there's no point ~ing", "it's no use ~ing". They follow a verb's て-form, and the two mean the same.
しょうがない is the everyday spoken form. It is a contraction of しようがない, "there is no way of doing it", which still turns up in writing. しかたがない is a touch more neutral and is the one you write, often as しかたがない仕方がない; in speech it drops が: しかたない. The polite forms are しょうがありません and しかたがありません.
With a passive verb, or about something the person brought on themselves, the same form says the result is deserved or only to be expected.
An い-adjective takes くても and a noun or な-adjective でも: "even if it is ~, nothing can be done about it".