What is the most "important" resonance structure of SCN⁻?
I performed a quick calculation on the DF-BP86/def2-SVP level of theory and analysed it with Natural Resonance Theory (from the Natural Bond Orbital Theory).
This results in the following major contributors to this wave function:
$$\left[
\underset{(1)}{\overset{67.49\%}{\ce{^-S-C#N}}} \ce{&lt;-&gt;}
\underset{(2)}{\overset{21.25\%}{\ce{S=C=N^-}}}
\right]$$

The third contributor is a weird structure with a 'long-distance-bond' between the sulfur and the nitrogen with $7.22\%$. All other contribution are neglected/discarded.

While your structure 3 is indeed a valid contributor, its actual contribution will be very small. Forcing the program to use it as a structure, it resulted in an error, as it was unable to match the orbitals to that structure. The reason for that is likely that the overlap between the sulfur and the carbon is too poor to actually be considered a good contributor. This would also explain the smaller contribution of the second structure.

Here are the localised (according to NBO) molecular orbitals:



(Colour code: blue/orange - occupied molecular orbital [Lewis];
red/yellow - virtual molecular orbital [unorccupied, non-Lewis])

The total contributions in terms of atomic orbitals to the above:


     (Occupancy)   Bond orbital / Coefficients / Hybrids
 ------------------ Lewis ------------------------------------------------------
   8. (1.98209) LP ( 1) S  1            s( 79.86%)p 0.25( 20.13%)d 0.00(  0.01%)
   9. (1.77474) LP ( 2) S  1            s(  0.00%)p 1.00( 99.91%)d 0.00(  0.09%)
  10. (1.77474) LP ( 3) S  1            s(  0.00%)p 1.00( 99.91%)d 0.00(  0.09%)
  11. (1.96342) LP ( 1) N  3            s( 52.42%)p 0.91( 47.54%)d 0.00(  0.04%)
  12. (1.99743) BD ( 1) S  1- C  2
               ( 44.86%)   0.6698* S  1 s( 20.64%)p 3.81( 78.61%)d 0.04(  0.75%)
               ( 55.14%)   0.7425* C  2 s( 51.41%)p 0.94( 48.47%)d 0.00(  0.12%)
  13. (1.99846) BD ( 1) C  2- N  3
               ( 41.02%)   0.6404* C  2 s( 48.44%)p 1.06( 51.48%)d 0.00(  0.08%)
               ( 58.98%)   0.7680* N  3 s( 48.02%)p 1.08( 51.75%)d 0.00(  0.23%)
  14. (1.99735) BD ( 2) C  2- N  3
               ( 43.98%)   0.6632* C  2 s(  0.00%)p 1.00( 99.91%)d 0.00(  0.09%)
               ( 56.02%)   0.7485* N  3 s(  0.00%)p 1.00( 99.80%)d 0.00(  0.20%)
  15. (1.99735) BD ( 3) C  2- N  3
               ( 43.98%)   0.6632* C  2 s(  0.00%)p 1.00( 99.91%)d 0.00(  0.09%)
               ( 56.02%)   0.7485* N  3 s(  0.00%)p 1.00( 99.80%)d 0.00(  0.20%)
 ---------------- non-Lewis ----------------------------------------------------
  16. (0.01904) BD*( 1) S  1- C  2
               ( 55.14%)   0.7425* S  1 s( 20.64%)p 3.81( 78.61%)d 0.04(  0.75%)
               ( 44.86%)  -0.6698* C  2 s( 51.41%)p 0.94( 48.47%)d 0.00(  0.12%)
  17. (0.01384) BD*( 1) C  2- N  3
               ( 58.98%)   0.7680* C  2 s( 48.44%)p 1.06( 51.48%)d 0.00(  0.08%)
               ( 41.02%)  -0.6404* N  3 s( 48.02%)p 1.08( 51.75%)d 0.00(  0.23%)
  18. (0.22011) BD*( 2) C  2- N  3
               ( 56.02%)   0.7485* C  2 s(  0.00%)p 1.00( 99.91%)d 0.00(  0.09%)
               ( 43.98%)  -0.6632* N  3 s(  0.00%)p 1.00( 99.80%)d 0.00(  0.20%)
  19. (0.22011) BD*( 3) C  2- N  3
               ( 56.02%)   0.7485* C  2 s(  0.00%)p 1.00( 99.91%)d 0.00(  0.09%)

               ( 43.98%)  -0.6632* N  3 s(  0.00%)p 1.00( 99.80%)d 0.00(  0.20%)




On the terminology.
Alchimista already explained most of this, however, I cannot stress enough: There is no such thing as a most stable resonance structure. Therefore when you say common, you probably mean large contribution to the wave function, and when you say rare, you probably mean little contribution. None of the resonance structures can be independent from each other, as they are all hypothetical. 

Please read more about that here: What is resonance, and are resonance structures real?