What classifies as a solvent/base in SN1, SN2, E1, and E2 reactions?
Okay in the question i am only seeing CH30H that simply means that methanol is acting both as a base and a solvent. 


Since methanol is acting both as a solvent and a reagent, it will undergo E1 reaction as you will know that it is a polar protic solvent and if the solvent was like **water(polar) or DMSO **  then it would follow SN1 and SN2 respectively and *CH30 (-) * would act as the nucleophile.  Thus solvents play an important role in understanding the type of reaction it would undergo.  So it is really important to understand the solvent. As per as i know there are no such conventions..( That  solvent is written below and reagent/base at the top).  If only one reagent  is given it is acting as a both solvent and a base. I would like to tell another thing that sometime we are mistaken a reagent to be a solvent : 
DMSO is often used as a oxidizing agent in presence of phosphoric acid.  We mostly confuse it and hold phosphoric acid as the oxidizing agent.