this is a very model-dependent question. In particular, you have not specified the representation (i.e. the charges) of the scalars you are looking for.
Very light particles at the lower end of the energy scale can also point towards the existence of new physics beyond standard model. ALPS experiment at DESY is searching for such new particles. Very Weakly Interacting Sub-eV Particles (WISP) naturally occur in various extensions of standard model. For example we can extend the standard model by including a light scalar which interacts very weakly with visible sector. But then it's mass and coupling with visible particles are (new) unknown parameters. WISP scenarios are supported by TeV-photon emission of active galactic nuclei. Please look at the ALPS website, http://alps.desy.de/ This experiment also looks for light axions.
For latest astrophysical bounds on sub-eV axion parameters see, https://arxiv.org/abs/1605.07668 and references therein.
Thanks for your in-depth answers. I asked this question because in the case of a new 'gauge boson', LHC limits its mass to be above 2-3 TeV, but at the same time community talk about a new MeV gauge boson.
I want to ask similar question for the scalar (pseudoscalar), like MeV scalar is allowed or not. I agree with you that it is a model dependent question, but If scalar does not couple to quarks then is it possible to take its mass in MeV range?
not necessarily. Even if you've got a pure singlet scalar, it will unavoidable couple to the Higgs and if that coupling is too large, there will be direct and indirect collider limits.
See "https://arxiv.org/abs/1501.02234" for a collection of such bounds.
Best regards,
Alexander
PS: Typically, if people talk about MeV-scales, these bosons have strongly suppressed couplings to the SM. Typically, they effectively operate within a dark sector, without really touching the SM.