Which of the following is the more stable carbocation?
Vinyl cations can also be generated by the protonation of acetylenes in strong acid.  The vinyl cations are captured by a nucleophile to produce a stable alkene with the nucleophile attached to the double bond.


  Which of the following is a more stable carbocation?


This route to vinyl cations has been well studied.  As shown in the following diagram, protonation of phenyl acetylene can proceed in two different ways.  The acetylenic carbon attached to the phenyl ring may be protonated or the terminal acetylenic carbon may be protonated.  As you can see, these two routes produce the two vinyl cations you want to compare.

Upon examining the product(s) found in this reaction, we find that only alkene A2 is produced.  Therefore, the reaction proceeds through the intermediacy of vinyl cation A1, not B1.  Consequently, the pathway involving A1 must be lower in energy, or said differently, vinyl cation A1 is lower in energy than vinyl cation B1.



Reasons Why Vinyl Cation A1 Is More Stable Than Vinyl Cation B1


The barrier to rotate the phenyl group is very low. Hence, in A1 the phenyl group can readily adopt a conformation where the aromatic p-orbitals are aligned with the empty p-orbital on the adjacent carbocation center. Additional resonance structures can now be drawn with the positive charge delocalized into the aromatic ring.  There is no such resonance stabilization with the aromatic ring possible for vinyl cation B1. This suggests that vinyl cation A1 is more stable than vinyl cation B1.
In addition to the resonance structures involving the aromatic ring, as shown at the bottom of the diagram, one can draw 2 more hyperconjugated resonance resonance structures for vinyl cation A1, but only 1 hyperconjugated resonance structure can be drawn for B1; again suggesting that vinyl cation A1 is more stable than vinyl cation B1.
Resonance effects are generally stronger and more controlling than inductive effects.  Therefore, the inductive arguments presented in @orthocresol 's answer play only a minor role, being outweighed by the stronger resonance effects described above.