事(こと)はないThere is no need to, Never happens

ことはない downplays the need for action and is often used to reassure, e.g. 焦ることはない ('There's no need to rush.') ことだ gives firm advice and carries an authoritative tone, e.g. 焦らずに進めることだ ('You should proceed calmly.') ことはない can also mean that something never occurs, which is a usage unrelated to ことだ, e.g. 彼と会うことはない ('I never see him.')

These two expressions can be opposite in meaning. ことはない means that something never happens or is not necessary, e.g. 彼と話すことはない ('There are no occasions where I talk to him'), while ことがある indicates occasional occurrence, e.g. 道に迷うことがある ('I sometimes get lost'). Note that ことはない can also express 'no need to', e.g. 心配することはない ('There's no need to worry'), which is unrelated to any use of ことがある.

These forms are opposites in meaning. ひつようがある shows something must be done, as in 行く必要がある ('You need to go.'). ことはない reassures that something doesn't need to be done, as in 行くことはない ('You don't need to go.'). One expresses necessity, the other expresses lack of it.

ことはない and すこしも～ない are both emphatic negatives, but with different nuances. ことはない means 'There's no need to (A).' or 'There are no instances of (A).'. For example, 心配することはない ('There's no need to worry.'), and 彼と話すことはない ('There are no occasions when I talk to him.'). In contrast, すこしも～ない emphasizes absence of degree or amount, e.g., すこしも心配していない ('I'm not even a little bit worried.').

ことはない has two primary uses: one is to express that there is no need to do something, e.g. 心配することはない ('There's no need to worry.'), and the other is to say that something simply doesn't happen, e.g. 彼と会うことはない ('There are no cases where I meet him.'). In contrast, はずがない expresses the speaker's strong denial or disbelief regarding some possibility, e.g. 負けるはずがない ('There's no way / It's highly unlikely we will lose.').

Both can serve as emphatic ways to express 'there is no need to', e.g. 心配する（までもない/ことはない） ('There is no need to worry.'). Note however that ことはない can also be used to mean that something never happens (lit. 'there are no cases of'), e.g. 彼と会うことはない ('I never meet him.'), while までもない is not used in this sense and will always have the 'no need' meaning.

Both express negation, but differ in tone and nuance. ことはない suggests that something never happens, or literally that 'there are no cases of (A)', e.g. 忘れることはない ('There's no way I'll forget'), while 決して〜ない adds strong emphasis, meaning 'never' or 'by no means', and could even be used together with ことはない, e.g. 決して忘れることはない ('I will by no means forget').

These expressions are similar in that both emphatically express negation, but they differ in nuance. ことはない implies something never happens, or literally 'there are no cases of (A)', e.g. 二度と会うことはない ('I'll never talk to him again / a second time.'). In contrast, わけがない expresses strong denial or impossibility, e.g. 負けるわけがない ('There's no way I can lose.')

ことはない and なくてもいい can overlap when expressing 'There is no need to...' or 'You don't have to...' For example, 心配することはない and 心配しなくてもいい both mean 'There's no need to worry.' However, ことはない can also be used to express that something never happens. For example, 彼と話すことはない means 'I never talk to him,' which is not a use for なくてもいい.