I am a single-member LLC being taxed as a disregarded entity. Can I depreciate a travel trailer. Are there any other tax deductions for the purchase and ongoing use?

"I am a corporately filed LLC".

No such thing. Either the business is incorporated as an S-Corp or C-Corp, or it is an LLC, the latter of which is a disregarded entity indicating you'll be reporting business income/expenses/assets on a SCH C as a part of your personal return.

If the business purchases the travel trailer that makes it a depreciable business asset. You still report personal use, and that personal use will affect your deductible expenses as well as depreciation, based on percentage of personal use.

If "you" purchase the travel trailer, it is NOT a business asset. Expenses for it's use will NOT be reported on SCH C. Instead, they get reported on the SCH A subject to the 10% AGI rule for itemizing your deductions. With this method, you'd claim your vehicle expenses in personal income under the Deductions & Credits tab in the Job Related Expenses section. Those expenses, along with all your other itemized deductions would need to be "at least" 10% of your total AGI in order to be deductible.

I suggest you utilize the first method and the business purchases the trailer.

Note that with either method, you MUST keep detailed records of your mileage when there is a mix between business use and personal use.

In some states, who purchases the vehicle (you or the business) doesn't matter. It's how you treat it that first year of ownership that determines if it's a business asset or not. For example, if you purchased the trailer years before you started your business, and wanted to make it a business asset, you can just "do it' that first year. But it's value as an asset will not be what you paid for it years ago when you purchased it. The business value for depreciation would be its FMV at the time you placed it in service for the business.