A study is made of the linear stability of finite‐thickness, anisotropic compressible shear layersv=zˆvz(x), with a parallel uniform magnetic field,B=zˆB0. The stability of such sheared flows, described by the double adiabatic equations of Chew, Goldberger, and Low [Proc. R. Soc. London Ser.  A  236, 112 (1956)], and involving the nonlocal coupling of the firehose and mirror modes caused by the velocity shear, is relevant in a number of astrophysical, geophysical, and space plasma configurations. The scalar perturbation quantities have the form  f (x) exp[i(kzz−&ohgr;t)]. The dimensionless variables characterizing a shear layer with a given velocity profile, assumed to be a linear profile in the present work, are the sonic Mach number,M ≡ (2vzm/S⊥), the ratio of the magnetic field energy density to the perpendicular thermal energy density,q2 ≡ (vA/S⊥)2, and the anisotropy parameter,r2 ≡ (S∥/S⊥)2. Here,vz(x=±∞)=±vzm,S∥, andS⊥are the sound speeds parallel and perpendicular to the magnetic field, andvAis the Alfve´n speed. The dimensionless variable characterizing the perturbation is the wavenumberB ≡ kzL, whereLis the shear layer thickness. It is shown that the resonance of the sound and firehose modes drives unstable standing and traveling waves for a shear layer having a vortex sheet profile [wherevz(x) is a step function]. For the vortex sheet, the unstable standing wave modes first appear atM=2(3)1/2rforr>(1+q2)1/2/2, and the unstable traveling wave modes first appear atM=0 forr<((1+q2)1/2/2). Numerical methods are used to generate values of &ohgr;rand &ohgr;i>0 (corresponding to unstable wave motion) for the ‘‘linear’’ shear layer in the (B, M) plane for various values ofqandr.The coverage of the (B, M) plane is forB≤5,M≤10, and for discrete values ofq≤0.5 andr≤2. Two regimes of instability are found to occur in the (B, M) plane with the structure of the unstable modes of the ‘‘linear’’ layer being very different from that for the anisotropic vortex sheet. The unstable modes are standing waves with &ohgr;r=0, and traveling waves with &ohgr;r≠0.