I do hear Entschuldigung a lot in this case, but what other words or phrases are there to politely excuse yourself while brushing past someone?
You can also say:
And you can put an “Oh, …” at the beginning of your excuse.
related: When to use 'Entschuldigen Sie'
You would use "Entschuldigen Sie (bitte), .." if you want something or just "Entschuldigung?" if it is clear what you want .. for example, if people block your way in the supermarket and you want to pass.
"Entschuldigung, dürfte ich kurz vorbei." would be the nicest. You can add that if the person is not getting what you want or if you want to be extra polite.
"Sorry" instead of "Entschuldigung" is shorter and very common today, but you would only use it if you are dealing with younger people.