Flautists seems to be very fond of Claude Debussy's Syrinx.
It was historically important as it brought solo flute music back from the dead. There don't seem to have been any important works for solo flute since the days of the Bach Family. Since Syrinx, however, solo flute works have become much more common. So flautists can be grateful to Debussy for that.
However, they're doubtless really fond of it because it's such a beguiling piece, short but full of dreams and sensuality and beauty. It's all melody, spinning out from the phrase heard at the start. Whether it depicts the nymph Syrinx being chased by the god Pan or the lament of Pan, it has a wonderfully exotic sound, thanks to its use of chromaticism and the whole tone scale as well as to its use of arabesques and incantatory repetition. Its phrases tend to fall at the end, like tears or sighs, culminating in the final diminuendo on a descending whole tone scale. This weight of expression expressed with French delicacy is another reason flautists love Syrinx. It's phrase structuring sounds so spontaneous, improvisatory even, but Debussy builds it in a long arc towards the climax and the closing dying away. It lasts less than three minutes, but packs a punch well above its weight.
Flautists tend to want to stamp their own individual responses on their performances of Syrinx. It's one of those works that needs to be heard performed by a range of performers. If you click on the following links you'll hear what I mean. No performance sounds alike.
Another famous classic of the 20th Century solo flute repetoire is Edgar Varèse's Density 21.5 (written for the platinum flute, the title refers to the specific gravity of the element). It lasts not much longer than the Debussy (around four minutes) and also packs a punch beyond its weight. Though Varèse seems to have sought to consciously distance himself from Debussy's Syrinx, it was his model and, whatever his intentions, it seems clear to me that Density 21.5 has much in common with Syrinx. You feel it straight away, in the quiet, mysterious opening, where a chromatic melodic cell is spun out into melody, with the same air of (illusory) spontaneity. It's a beautiful opening. Later though, Varèse veers towards atonality, making much of the tritone (the devil in music) and taking chromaticism much further than Debussy. Moreover, the intervals in his melodic lines are prone to much wider leaps, generally (but not always) upwards, for, whereas the phrase in the Debussy piece tended to fall, those of Density 21.5 tend to rise, aspire, soar. Like the Debussy though, it has a definite overall shape - crudely put, a journey from darkness to light. I'm very fond of the piece, though when I first heard it (many years ago) I didn't get it. Tastes can change.
There are a number of performances of Density 21.5 on YouTube. They are all individual, of course, but this composer leaves them much less space for expressive license:
Finally, if you enjoy Syrinx and Density 21.5, it's likely you'll warm to Air by everyone's favourite Japanese classical composer, the late Toru Takemitsu. Takemitsu composed many quiet, reflective works reflecting his country's landscapes and gardens, drawing on the music of Debussy and Messiaen above all, though not without a little serial input, to express his deeply poetic worldview. The results can be very attractive. Air is a solo work flute that clearly owes something to Syrinx and I suspect also to Density 21.5. Chromaticism, touches of the whole tone scale, modal writing (after the manner of Messiaen), all inform his melodic writing, with little melodic hooks drawing the listener in. You can listen to it here. I hope you do!