Singular offers no syntax to denote inequalities. So the answer is almost no,
with the the exception of linear equations.
This is, the simplex
method is available: http://www.singular.uni-kl.de/Manual/la ... htm#SEC426
To answer the question in a wider range about real algebraic computations,
Singular offers some libraries: realrad_lib
for Computation of real radicals http://www.singular.uni-kl.de/Manual/la ... tm#SEC1336
Two libraries for Counting the number of real roots of uni- and multivariate polynomial systems
rootsur_lib http://www.singular.uni-kl.de/Manual/la ... tm#SEC1918
and rootsmr_lib http://www.singular.uni-kl.de/Manual/la ... tm#SEC1902
These implement:Descartes' rule of signs, the Budan-Fourier theorem, Sturm sequences and Sturm-Habicht sequences
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:David J. Wilson, Russell J. Bradford, James H. Davenport
Speeding up Cylindrical Algebraic Decomposition by Gröbner Baseshttp://arxiv.org/abs/1205.6285
(Though, this work reports on calculations with Maple.)