We study plasma instabilities in inductive discharges using SF6 and Ar/SF6 gases. Instabilities occur in transition from capacitive to inductive mode of the discharge . The capacitive coupling plays a crucial role in the instability process. A variable electrostatic (Faraday) shield has been used to control the capacitive coupling from the excitation coil to the plasma. An increase of the shielded area reduces the capacitive coupling and leads to the reduction of stable capacitive and unstable regions of the discharge. The plasma instability disappears when the shielded area exceeds 65 percent of the total area of the coil. A global model of instability gives a slightly higher value of 85 percent for instability suppression with the same discharge conditions (Ar/SF6 1:1, 5 mTorr). The variations of plasma parameters have been explored for Ar/SF6 gas pressures between 5 and 40 mTorr. Space and time variations of plasma density have been observed at higher pressures during the instability. The instability decays toward the center of the chamber more rapidly as the gas pressure increases. Weaker plasma instabilities have been also observed in pure inductive discharges for an Ar/SF6 gas pressure above 20 mTorr. This weaker instability, not related to the E-H transition, is probably an attachment instability. Measurements of the chemical composition of the plasma have allowed improvements in the instability model.