We check the shape of each species using VSEPR / known molecular geometry.

Analyze one by one:

(i) $\mathrm{XeF_2}$

• Central atom: Xe
• Xe has 5 electron pairs around it: $2$ bond pairs + $3$ lone pairs
• Electron pair geometry: trigonal bipyramidal
• Lone pairs occupy equatorial positions, so the molecular shape is linear.

(ii) $\mathrm{I_3^+}$

• Total valence electrons $= 3\times 7 -1 = 20$
• Structure is analogous to a central I bonded to two I atoms with $2$ lone pairs on central atom
• Steric number on central I $=4$
• Electron pair geometry: tetrahedral
• Molecular shape: bent (V-shaped), not linear.

(iii) $\mathrm{C_3O_2}$

• Carbon suboxide structure: $\mathrm{O=C=C=C=O}$
• Each central carbon is $sp$ hybridized
• Entire molecule is linear.

(iv) $\mathrm{I_3^-}$

• Central I has $2$ bond pairs and $3$ lone pairs
• Trigonal bipyramidal electron arrangement
• Molecular shape: linear.

(v) $\mathrm{CO_2}$

• Central C has two electron domains
• $sp$ hybridized
• Shape: linear.

(vi) $\mathrm{SO_2}$

• Central S has three electron domains ($2$ bonds + $1$ lone pair)
• Trigonal planar electron arrangement
• Molecular shape: bent, not linear.

(vii) $\mathrm{BeCl_2}$

• Central Be has two bond pairs and no lone pair
• $sp$ hybridized
• Shape: linear.

(viii) $\mathrm{BCl_2^-}$

• Valence electrons: $3 + 2\times 7 + 1 = 18$
• Central B has two bond pairs and one lone pair
• Steric number $=3$
• Electron pair geometry: trigonal planar
• Molecular shape: bent, not linear.

Linear species are:

• $\mathrm{XeF_2}$
• $\mathrm{C_3O_2}$
• $\mathrm{I_3^-}$
• $\mathrm{CO_2}$
• $\mathrm{BeCl_2}$

Total number of linear species: $5$