Do vinyl cations adopt a classical or non-classical structure?
TL;DR: there have been many theoretical investigations of the relative energies of these two forms for the parent vinyl cation with more recent work indicating that the bridged form is slightly more stable (by about 1–3 kcal/mol)4,5. This prediction has received support from recent experimental work as well6.

Taking such relatively small energy differences into account I would not be surprised if the situation is reversed with a particular substituent. Thus, answering the second part of the question ("Does it depends on what the substituents are?") requires high-level calculations for a wide variety of different substituents which seems to be absent in literature yet.



Classical structure: trigonal (sp2) or linear (sp)?

First note that (give Marko a credit for paying our attention to that), as it is mentioned in Stang's "Vinyl Cations" book, the bent $\mathrm{sp^2}$-hybridized structure with a $\ce{C=C-H}$ bond angle of 120° (OP calls it "classical") is about 50 kcal/mol higher in energy than the linear $\mathrm{sp}$-hybridized form in all of the theoretical calculations. 


  These calculations confirm the organic chemist's intuitive and
  qualitative feeling that carbenium ions prefer a trigonal planar
  geometry with an empty p orbital and that it takes energy to bend the
  methyl cation from its low-energy planar form.


Thus, in the studies below the meaning of the classical structure is different from the OP's one: the linear $\mathrm{sp}$-hybridized structure is called classical in almost all the studies.





So, Stang's "Vinyl Cations" references a CI/DZP study of the vinyl cation by Weber, Yoshimine, and McLean 1 as the most recent ab initio correlated calculations published at the time the book was written (the late 70s), so I'll start a review from this study.

Weber, Yoshimine, McLean, 1976

It is was found that the linear and bridged conformations have energies equal to within 0.01 kcal/mol with the later being lower in energy. However, such a small difference is below the accuracy of the CI/DZP level of theory, and thus, the authors didn't claim that the bridged conformation is more energetically favourable, just that it can be such ("probably" is the key word in the quote from the paper below).


  While our CI calculation shows the linear and bridged conformations to
  have energies equal to within 0.01 kcal/mol, we clearly cannot claim
  this absolute accuracy. In fact, taking into account basis set
  deficiencies, both in the one- and $n$-particle spaces, along with
  optimization effects as discussed above, we interpret our calculations
  to predict the relative energies of the two structures to be within
  1–2 kcal/mol with the bridged structure probably having the lower
  energy.


Besides, note that in the study the geometries have been optimized at the SCF level rather than at the correlated while CI calculations were done at the SCF optimized geometries. Taking into account that correlation effect were found to have major impact on relative stability of the two forms as mentioned in the study, it is especially interesting to look at the results obtained by Raghavachari, Whiteside, Pople, and Schleyer2 who optimized the structures not only at the HF/6-31G* but also at the MP2/6-31G* levels as well as did more high-level correlated calculations (MP4) with a bigger basis set (up to 6-311G** of TZP quality).

Raghavachari, Whiteside, Pople, Schleyer, 1981

For the bridged structure the $\ce{C-C}$ bond length was found to increase from 1.21 Å at HF/6-31G* to 1.23 Å at MP2/6-31G* while the distance of the bridging hydrogen to the middle of the $\ce{C-C}$ bond stays the same (1.18 Å). For the classical structure the authors report no significant changes in the geometry. With respect to the relative stability it was found that use of the MP2 geometries results in a slight additional stabilization to the bridged form: the MP4(SDQ)/6-31G**//MP2/6-31G* energy difference is 1.3 kcal/mol while the MP4(SDQ)/6-31G**//HF/6-31G* energy difference is only 0.6 kcal/mol (in both cases in favour of the bridged form).


  With the larger 6-311G** basis, the bridged structure is stabilized
  further, and our final result at MP4(SDQ)/6-311G**//MP2/6-31G*
  indicates 11 [bridged] to be more stable than 10 [classical] by 3.0 kcal/mol.


The authors also mention that these final results are quite in line with the CEPA study by Lischka and Kohler3 done using a comparable basis set: the Lischka and Kohler geometries were very similar to the HF/6-31G* geometries obtained in the study and the bridged form was found to be 4.0 kcal/mol lower in energy than the classical form as well.

A further study done by the Pople group is also is in agreement with the findings quoted above: the bridged form predicted to be more stable by 3.1 kcal/mol (including zero point energies)4. А study by Lee and Schaefer5, though, reports a somewhat lower difference. 

Lee, Schaefer, 1986

In a recent CI/DZP study by Lee and Schaefer with the largest CI expansion considered being CISDT the energy difference was found to be 0.68 kcal/mol with the nonclassical structure again lying lower. The authors also considered the effect of the zero-point vibrational energy corrections and found that it further lowers the nonclassical structure by 0.29 kcal/mol.


  Our best strictly ab initio estimate of $\Delta E$(classical-non- classical) for $\ce{C2H3+}$ is obtained by adding the above 0.29 kcal correction to the 0.68 kcal difference from CISDT. In this manner, $\Delta E$ is predicted to be 0.97 kcal/mol. We estimate that a proper variational treatment of quadruple excitations might increase this $\Delta E$ value by perhaps 0.5 kcal/mol.





Weber, J.; Yoshimine, M.; McLean, A. D. A CI study of the classical and nonclassical structures of the vinyl cation and their optimum path for rearrangement. J. Chem. Phys. 1976, 64 (10), 4159–4164. DOI: 10.1063/1.431986.
Raghavachari, K.; Whiteside, R. A.; Pople, J. A.; Schleyer, P. V. R. Molecular orbital theory of the electronic structure of organic molecules. 40. structures and energies of C1-C3 carbocations including effects of electron correlation. J. Am. Chem. Soc. 1981, 103 (19), 5649–5657. DOI: 10.1021/ja00409a004.
Lischka, H.; Koehler, H. J. Structure and stability of the carbocations C2H3+ and C2H4X+, X = hydrogen, fluorine, chlorine, and methyl. Ab initio investigation including electron correlation and a comparison with MINDO/3 results. J. Am. Chem. Soc. 1978, 100 (17), 5297–5305. DOI: 10.1021/ja00485a010.
Curtiss, L. A.; Pople, J. A. Theoretical study of structures and energies of acetylene, ethylene, and vinyl radical and cation. J. Chem. Phys. 1988, 88 (12), 7405–7409. DOI: 10.1063/1.454303.
Lee, T. J.; Schaefer, H. F. The classical and nonclassical forms of protonated acetylene, C2H3+. structures, vibrational frequencies, and infrared intensities from explicitly correlated wave functions. J. Chem. Phys. 1986, 85 (6), 3437–3443. DOI: 10.1063/1.451828.
Crofton, M. W.; Jagod, M.; Rehfuss, B. D.; Oka, T. Infrared spectroscopy of carbo‐ions. v. classical vs nonclassical structure of protonated acetylene c 2 h + 3. J. Chem. Phys. 1989, 91 (9), 5139–5153. DOI: 10.1063/1.457612.