The main issue for single-member LLCs in the US is to actually operate it as a separate entity. Don't share assets, accounts, etc. Don't personally guarantee debt unless you absolutely have to (and then you'll be liable for that). And of course you're still liable if you personally break the law. If you hire an employee and they assault a customer or something, your company may be sued like any larger company, but they probably won't be able to pierce the veil and go after you personally to collect.
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The main issue for single-member LLCs in the US is to actually operate it as a separate entity. Don't share assets, accounts, etc. Don't personally guarantee debt unless you absolutely have to (and then you'll be liable for that). And of course you're still liable if you personally break the law. If you hire an employee and they assault a customer or something, your company may be sued like any larger company, but they probably won't be able to pierce the veil and go after you personally to collect.