The right spelling is Дмитрий Дмитриевич Шостакович. Or Дмитрий Шостакович if omitting the patronymic.
He was Russian.
Transliteration of his name into the Latin alphabet varies:
Dmitri Dmitriyevich Shostakovich (English transliteration),
Dmitri Dmitrijewitsch Schostakowitsch (German transliteration),
Dmitri Dmitrievitch Chostakovitch (French transliteration),
Dmitrij Dmitrievič Šostakovič (scientific transliteration),
and so on.
Adding: To copy (and edit) this from my comment below before an admin comes and deletes them all, Dimitri is a cognate of Dmitrij. (Cognate: a variant of a word evolved from the same older form, usually in a different language.) Replacing people's names with their cognates more familiar to the audience used to be common practice, for example you've probably heard of the writer Leo Tolstoy, whose name was actually Lev (Лев), or you might have come across another composer named as Peter Ilyich Tchaikovsky, rather than directly transliterated Pyotr (Пётр). Nowadays this practice is on its way out and people's names are more often given in their original foreign form.