They go on your taxes for the year the payment is included on your 1099. If the check is in the mail, odds are it will be on the 1099 they issue you in a few weeks. Thus, you have to include it on this year's taxes, because the government will get a record of how much you were paid, and will be none too pleased if you report a different amount.
Assuming the OP follows cash basis accounting rather than accrual (pretty common for sole proprietors) this isn't actually true. The income should be applied to the year in which it has been constructively received which mostly means when the check is in your possession. If the employer is local it can also include when the payment was available for you to pick up whether or not you actually did. Here's some more info:
https://www.law.cornell.edu/cfr/text/26/1.451-2From a practical perspective, I'm sure you're not going to get in trouble if you just follow what the 1099 they issue says since, as iamlindoro says, the IRS will receive this record and will have an issue if the discrepancy isn't explained. If you do want to claim the income in 2016 instead despite it being reflected on the 1099 for 2015, the recommendation I've seen is that you should claim the income but add an expense for the same amount saying something to the effect of "income not constructively received in 2015." You will of course need to make sure you include this income on your 2016 taxes.
Something else you might want to be a little wary of is the postmark. If it was mailed a few days before the end of the year, I would think you'd be safe (I'd keep the envelope), but if it was mailed awhile ago and it's believable that you would have received it by the end of the year I would be a little more cautious.
Too late now, but this year I stopped sending invoices after the middle of December and if questioned explained I'd be fine sending one if they could commit to getting the check to me by the end of the year, but otherwise I'd rather wait to avoid this discrepancy. Didn't seem to be a problem, though most of my invoices go out at the end of the month anyway.