Singular

Singular offers no syntax to denote inequalities. So the answer is almost no,
with the the exception of linear equations.
This is, the [b]simplex[/b] method is available: http://www.singular.uni-kl.de/Manual/latest/sing_386.htm#SEC426

To answer the question in a wider range about real algebraic computations,
Singular offers some libraries:
[b] realrad_lib [/b]for [i]Computation of real radicals [/i]
http://www.singular.uni-kl.de/Manual/latest/sing_1260.htm#SEC1336

Two libraries for
[i]Counting the number of real roots of uni- and multivariate polynomial systems
[/i]
[b]rootsur_lib[/b] http://www.singular.uni-kl.de/Manual/latest/sing_1842.htm#SEC1918
and [b]rootsmr_lib [/b] http://www.singular.uni-kl.de/Manual/latest/sing_1826.htm#SEC1902 respectively.

These implement:
[i]Descartes' rule of signs, the Budan-Fourier theorem, Sturm sequences and Sturm-Habicht sequences[/i].

The computation of the Sturm-Habicht sequences (subresultants) are imititated by Groebner basis calculations.
Hence it does not perform as well as a proper implementation could do.
(Although subresultants are implemented in the kernel of the factorization engines,
these commands are not directly accessible to the user.)

For those interested in usage of Groebner basis as a preprocces in semialgebraic computations,
see for instance:

[i]David J. Wilson, Russell J. Bradford, James H. Davenport
Speeding up Cylindrical Algebraic Decomposition by Gröbner Bases[/i]
http://arxiv.org/abs/1205.6285 (Though, this work reports on calculations with Maple.)

Is there any library able to solve semialgebraic polynomial systems? Or, at least, dealing with regular chains?

Thanks.