According to the Tax Foundation reading of the matter, in the final conference report (as just passed in the Senate, with a slightly altered name), there is an explicit exclusion to prepaying State and Local Taxes, except for property taxes, in 2017 for 2018's taxes.
This would mean, for example, that you cannot make an estimated income tax payment for 2018, and deduct it on your 2017 taxes.
However, property taxes are apparently excluded from this. See the conference report, about page 88:
SEC. 11042. LIMITATION ON DEDUCTION FOR STATE AND LOCAL, ETC. TAXES.
(a) In General.--Subsection (b) of section 164 is amended by adding at the end the following new paragraph:
``(6) Limitation on individual deductions for
taxable years 2018 through 2025.--In the case of an
individual and a taxable year beginning after December
31, 2017, and before January 1, 2026--
``(A) foreign real property taxes shall not
be taken into account under subsection (a)(1),
and
``(B) the aggregate amount of taxes taken
into account under paragraphs (1), (2), and (3)
of subsection (a) and paragraph (5) of this
subsection for any taxable year shall not
exceed $10,000 ($5,000 in the case of a married
individual filing a separate return).
The preceding sentence shall not apply to any foreign
taxes described in subsection (a)(3) or to any taxes
described in paragraph (1) and (2) of subsection (a)
which are paid or accrued in carrying on a trade or
business or an activity described in section 212. For
purposes of subparagraph (B), an amount paid in a
taxable year beginning before January 1, 2018, with
respect to a State or local income tax imposed for a
taxable year beginning after December 31, 2017, shall
be treated as paid on the last day of the taxable year
for which such tax is so imposed.''.
(b) Effective Date.--The amendment made by this section shall apply to taxable years beginning after December 31, 2016.
Notice this specific detail:
For purposes of subparagraph (B), an amount paid in a
taxable year beginning before January 1, 2018, with
respect to a State or local income tax imposed for a
taxable year beginning after December 31, 2017, shall
be treated as paid on the last day of the taxable year
for which such tax is so imposed.
That only applies to income taxes. Of course, it effectively applies to sales/use taxes, insomuch as you can't typically "prepay" them before you actually owe them; I suppose there could be some way to shenanigan this, but most likely the one real way to benefit in the 2017 tax year is to make your purchase in 2017. So if you're deducting sales taxes, and planning to buy a car soon, perhaps do so in 2017.
Property taxes, however, can be prepaid to your heart's content - especially in states, like mine (Illinois) which actually bill you 2017's taxes in 2018 - thus allowing you to benefit from itemized deductions and then take a standard deduction next year.
I second user102008's recommendation to also pay any actually-owed state income taxes for 2017 prior to the end of 2017; if you're not expecting a refund, it's a smart choice.