様(よう)がないThere's no way to, It's impossible to

These both express impossibility, but in different ways. はずがない asserts a strong disbelief based on knowledge, like 雨が降るはずがない ('It can't possibly rain.'). ようがない・ようもない focuses on the lack of a method, as in 連絡のしようがない ('There's no way to contact them.').

わけがない and ようがない・ようもない are similar in that both express impossibility. わけがない rejects something as logically unthinkable, e.g. 彼がそんなことをするわけがない ('There's no way he'd do that'). ようがない emphasizes that there is no way or means to do something, e.g. 助けようがない ('There's no way to help').

While both express impossibility, そうもない focuses on visual or situational cues suggesting unlikelihood, e.g. '雨が降りそうもない' ('It doesn't look like it'll rain'). ようがない・ようもない implies there is no method or way to perform the action, e.g. '連絡のしようがない' ('There's no way to contact them').

Both ようがない・ようもない and がたい express difficulty or unfeasibility of an action, but differ in nuance. ようがない・ようもない conveys that there is no way or means to do something. For example, 連絡のしようがない ('There's no way to contact him'). がたい, on the other hand, indicates that something is very difficult to do, often due to factors beyond the speaker's control. For example, 信じがたい話だ ('It's a story that is hard to believe').