I'm not sure if this idea can be found anywhere else on the forum. If so, please tell me. My idea is to give your favourite melodies played by a solo instrument which brings out the character of the instrument. Here are some of my ideas:

Cello

Sibelius - 1st theme 1st Movement Fourth Symphony
Elgar - Main Themes 1st and 3rd Movements Cello Concerto
Shostakovich - Slow movement 15th Symphony

Oboe

Samuel Barber wrote at least 2 amazing melodies for oboe. He seems to have had a special affinity for this instrument:

Symphony No 1 - Tune from 3rd slow section
Violin Concerto - 1st theme from slow movement

Just off the top of my head:

Clarinet:
The opening clarinet of Gershwin's Rhapsody in Blue-- not one of my most favourite compositions per se, but its so unforgettable. The opening glissando is downright kinky.

English horn:
Sibelius' Swan of Tuonela.

Flute:
Certainly Debussy's Afternoon of the Faun-- actually many of the woodwind parts in that piece are amazing. The flute in that lower register brings out the more velvet-like tones.

Oboe:
The sad, sad oboe solo that trails off at the end of Takemitsu's Tree-Line. There are some slight pitch bends that occur that give it a particularly "eastern" sound.

Bassoon:
Of course, the opening of Stravinsky's Rite of Spring. I wonder have there been any other composers that have used the bassoon in such an unusually high register?

To say nothing of Bach's solo violin and cello works!

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At last to guess, instead of always knowing. To be able to say “ah” and “oh” and “hey” instead of “yea” and “amen. ” ~ Wings of Desire

French horn:

Tchaikovsky 5th and Mahler 5th themes. This is what french horn was made for, despite all those concertos and wind ensamble works that put him, among the bassoon, on list of most silly sounding instruments in music.

Flute:

Slow movement of Ravel concerto in G.

Trombone:

Tchaikovsky 3rd symphony, 4th movement, mystical trombone solo.

Cello:

Elgar concerto, first movement theme. It's not my favuorite concerto for cello, but when we are talking about "bringing out the instrument's character", where can I find more weeping sould of cello? And cello, for me, is weeping instrument.

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English Horn:
Largo from Dvorak Symphony No. 9, best key ever too.

Flute:
Poem for Flute and Orchestra by Griffes, I don't think I've heard (and performed) a more profound work for flute. The light-hearted flute be profound? Yes, with all the anger, anxiety, pain, and sadness one could find from any other instrument.

Clarinet:
3rd movement from Rachmaninoff Symphony No. 2, completely ruins my nerves to hear it played.

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Criticism is prejudice made plausible. - H. L. Mencken