There is no "list." You decide what pieces you feel are most powerful to you through many years of listening and study. There are, however, a number of pieces that you are expected to know as a student of composition at any serious music school - walking into a music school without having a thorough knowledge of "the basics" - the oeuvres of composers like Bach, Beethoven, Mozart, Brahms, etc. - would be like attempting to attain an English degree without any knowledge of Shakespeare.
Unfortunately (or perhaps fortunately, in the end) for you, you can't get by in a serious pursuit of composition without a knowledge of and ability to appreciate modern and contemporary works, either (if we were to continue the literary analogy here, trying to study composition without knowing some of the important works by Ligeti, Messiaen or Cage and others would be something akin to trying to obtain a degree in literature without exposure to Beckett, or Joyce.)
As this latter part of the repertoire seems to be a stumbling block for you, it may prove beneficial to have a list of works that might serve as "more accessible", yet still important introductions to the bodies of work produced by some of these composers. Here are a few:
Cage: Sonata XIII from Sonatas and Interludes - Something akin to a lullaby that a music-box might produce; very lovely writing.
Bartók: Concerto for Orchestra - a cornerstone of the 20th century orchestral repertoire.
Debussy: La Mer - a cross between a tone-poem and symphony, one of the most important works by this composer.
Ligeti: Musica Ricercata - a series of pieces for solo piano, mostly studies in the integration of folk music into what was becoming Ligeti's language at this time (although this was not necessarily his intention) - interestingly organized in that each piece in the set uses one more pitch-class than the last, culminating in a usage of all twelve pitches of the chromatic scale. One of these pieces is a funeral dirge in memory of Bartók, with sonorities that suggest booming church bells.
Stravinsky: Symphony of Psalms - as the titles suggests, a work for orchestra and choir after texts from the Psalms, sung in Latin.
Messiaen: "Praise to the Immortality of Jesus" from Quartet for the End of Time - a singularly gorgeous meditation for violin and piano, wherein the violinist is given the somewhat strange expressive indication to play the part in a "paradisiac" fashion.