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Nucleophilicity is highest in polar aprotic solvents, and lower in polar protic solvents because polar protic solvents (like water) can make H-bond and it will ultimately make the nucleophile more bulky.
E.g. of polar protic solvents - H2O, CH3OH, CH3CH2OH, CH3COOH, H2SO4
E.g. of polar protic solvents - DMSO, Acetone, DMF, HMPA, Acetonitrile
If the solvent is polar aprotic, it’s safe to assume the reaction is SN2. If the solvent is polar protic, the reaction will likely be E2.
Solvent doesn't help in deciding about SN1 or E1, you have to go through the concept for solving this.