1. Hope your wishes all come true.

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Is "I" omitted before "Hope" in this sentence? Why is this sentence used without the subject "I"? In which situation is this sentence used?

Eliminating the "I" is very informal, bordering on slang usage, certainly colloquial. This would probably not be done in spoken language, but in a terse/quick inscription in a greeting card, for example, below the printed verse, with the signature. The signature indicates who is hoping instead of writing "I hope.."