These are not all traditional "musicals." I have also included movies for which music is a central concern or element, like biographies of or fictional stories about musicians. After each title is a note describing the type of music featured in the movie and other explanations where appropriate, in addition to my star rating for the film.
(NOTE: Dates covered are mid-1983 through 1996. I didn't start using stars until late 1989, so ratings for movies older than that have been added after the fact. Also, I did not write as many reviews as Siskel & Ebert. These are definitely NOT the only musical movies, or the best musical movies, to come out during this period; they're just the ones I have reviews for.)
ALADDIN. Traditional musical, but animated. 
AMADEUS. Classical; fictionalized biography of Mozart. 
BEAUTY AND THE BEAST. Traditional musical, but animated.
DOORS, THE. Rock. 
EDDIE AND THE CRUISERS. Rock; fictional band. 
FABULOUS BAKER BOYS, THE. Pop standards, lounge music.
HEARTBREAK HOTEL. Rock; fictionalized episode about Elvis.
LA BAMBA. 1950's rock.
LITTLE MERMAID, THE. Traditional musical, but animated.
LITTLE SHOP OF HORRORS, THE. Traditional musical in form, if not in content.
MR. HOLLAND'S OPUS. Variety of types of music; story of high school band director.
NIGHTMARE BEFORE CHRISTMAS. Traditional musical, but animated.
SISTER ACT. Oldies girl-group rock.
STAYING ALIVE. Disco dancing, Broadway music (original, Saturday Night Fever, was better).
SWEET DREAMS. Country; biography of Patsy Cline.
TAP. Dancing (tap, of course)
UNFAITHFULLY YOURS. Classical; story of symphony musicians.
WHITE KNIGHTS. Ballet.