There seem to be some problems with the Standard Model, but many beyond the standard model theories allow or predict phenomena that are not well motivated experimentally. One might define a "constrained" beyond the Standard Model theory as one that lacks any of a number of common flaws of such models. Some constraints that might be particularly relevant are that constrained theories predict that:
1. Baryon number is separately conserved.
2. Lepton number is separately conserved. This implies, among other things, that neutrinos have Dirac mass.
3. Neutrinoless double beta decay is not possible.
4. Proton decay is not possible.
5. Flavor changing neutral currents do not exist.
6. Net electromagnetic charge is conserved.
7. Combined CPT symmetry is observed. This implies among other things that particles and antiparticles have equal rest masses.
8. Combined mass-energy is conserved.
9. There are no magnetic monopoles.
10. The strong force does not give rise to CP violations.
11. Photons are massless.
12. "The W is a quite democratic particle: it decays with equal frequency into all the possible lepton pairs and quark pairs that are energetically allowed. Electron-electron antineutrino, muon-muon antineutrino, and tau-tau antineutrino pairs are equally likely; down-antiup and strange-anticharm quark pairs are three times more likely than the lepton pairs, because they exist in three colour-anticolour combinations." This implies that experiments can impose very strict limits on so far undiscovered quarks and leptons, essentially ruling out any new quarks or leptons with lower masses than the third generation of quarks and leptons in the Standard Model, and also probably ruling out novel classes of fermions (at least at spin 1/2), or models with more than three color charges for quarks. More deeply, it rules out any model that disrupts the fundamental equivalency in some sense of lepton and quark types shown by W decay patterns.
Notes On SM4 Limits
Incidentally, some of the precision electroweak data lower bounds on fourth generation lepton masses (from LEP II as of 2007) are as follows: "A robust lower bound on fourth–generation masses comes from LEP II. The bound on unstable charged leptons is 101 GeV, while the bound on unstable neutral Dirac neutrinos is (101, 102, 90) GeV for the decay modes ν4 → (e, μ, τ) + W. These limits are weakened only by about 10 GeV when the neutrino has a Majorana mass."
For comparison, the Tau (which is the heaviest known unstable charged lepton) has a mass of about 1.8 GeV (a bit less that 1/50th of the precision low bound) and the Tau neutrino mass is probably easily a thousand times less than the Tau and probably closer to a million times less than the Tau. The lower limits on neutrino mass, thus, seem stringent, but not insurmountable given some evidence from Neutel 2011 papers that there are more than three generations of neutrinos from more than one type of experiment.
Limits on fourth generation quark masses (256 GeV for t' and 128 GeV for b' as of 2009) were much closer to the third generation quark masses, and hence not very constraining, although the limits imposed by experiment on CKM matrix values in a fourth generation standard model are quite strict.