We already know when a colon should be used, but when should a semicolon or dash be?

Semicolon

You use a semicolon when you are connecting two independent clauses that link because of similar ideas, or when introducing a list that is more complicated because it has explanations. Here are two examples of how semicolons are most commonly used:

• I like to go to the library; there are always so many excellent books there.
• You should bring a toothpaste for hygiene; a bottle of water for hydration; a waterproof coat in case it rains; and a snack to eat whilst en route.

Dashes

You can use dashes (more commonly known as em dashes in this style of writing) in the place of parentheses so it doesn’t make for an odd end to a sentence, or you can use them to replace a colon in more casual writing. Like this:

• He knew there was only one thing for it, so he ran – or rather, scurried. – We could have written this sentence as He knew there was only one thing for it, so he ran (or rather, scurried). But the dash looks better, even though both are correct.
• Sam liked three things – reading, writing, and video games. – Notice how this replaces the colon in the example from earlier.