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Beyond that, many amateur and even some non-euphonium-playing professional players are not clear whether a euphonium is pitched in Bb or in C. Most publications that include euphonium parts, which may be called either "Euphonium" or "Baritone Horn," include both a treble clef and bass clef part.
The treble clef part is in transposed treble clef. Because a euphonium's open fundamental note sounds as a B-flat, and because we are in the mid-low octave among brass, the part is transposed up an octave and a whole step to make up for the instrument sounding an octave and a whole step down. A treble clef middle C on the first ledger line below the staff, actually sounds as a Bb on the 2nd bass clef line. We could share a part with tenor sax or bass clarinet and be in total agreement on our notes.
The bass clef part is non-transposed, including the printed octave. A written middle C on the first ledger line above the staff sounds as a C in the middle-C octave. We could share a bass clef part with a trombone or bassoon and be in total agreement on our notes.
Therein lies the confusion. So armed with the description above, you can try to answer the question for anyone who asks (and good luck on that!).
A little more background...
In the typical British brass band you will find transposed treble clef parts for ALL instruments except the bass trombone. Trombones 1 & 2 are in Bb treble clef. And there are separate parts for the Eb and BBb tubas, transposed for each instrument.
USA grade schools might usually start a euphonium player on bass clef. However, many times a trumpet player will be switched to euphonium, in which case treble clef music makes the most sense. Colleges seem to prefer bass clef euphonium parts, perhaps because the instructors there have usually been either tuba or trombone players.
Sometimes in a smaller band a euphonium player may need to play the part for a missing instrument. Bassoon, trombone or even tuba parts could be comfortable for a bass clef player, while bass clarinet or tenor saxophone parts would be comfortable for a treble clef reader.
I have observed that a very few modern publications may have only bass clef parts. That is becoming less common with more music being input on computer, because a couple of clicks can produce the treble/bass parts. In some older publications I've seen only treble clef parts, and brass band music would be only in treble clef unless it has been adapted to fit players of mixed background.
In some French music you may even find Bb BASS CLEF parts! Those can be very tricky for bass clef players from the USA.
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What follows is my strongly-held opinion, and many euphonium players (and other musicians) will disagree.
I believe treble is the best clef for euphonium.
First, when a piano reads music in C, it is using the white keys. That same logic would say our scale that starts on our open pitch should be called a C scale, which it is in treble clef. To me, that helps create a logical, and subconscious, relationship with the instrument. A price we pay for that is not "talking" in concert pitch, but we share that burden with many other instruments. Most band/orchestra conductors are used to talking to the general group in concert pitch and it's our job to know that when the boss says "E-flat" we need to think of our treble clef F. (Many conductors talk to a specific instrumental section in written pitch, as in "your 'F' at measure 12.")
Second, in MOST standard euphonium music we play, a treble clef part will need fewer ledger lines. For treble parts, it gets visually intense around E above the staff or F below the staff. Within that range are 99% of the notes we'll play most days. (In bass clef the visuals get intense around G above the staff and pedal A below the staff. So the E in treble I just mentioned would be a D in bass clef, needing 5 ledger lines!)
Third, the treble clef transposition standard is in keeping with many band/orchestra instruments. The entire clarinet and saxophone families follow the transposed standard, for example. And a Db piccolo will get a Db part and an Eb horn will get an Eb part (at least in older music, where these instruments were more common). Trumpets usually use this as well, to accommodate Bb, C, and Eb trumpets. French horns (I know, I know...they are just called "horns" now) are F instruments with an F part; the part is mostly in treble clef, but when it does into bass clef it is still transposed into F. See the chart below for examples of how a concert middle C on the piano is written for various instruments.
More on point 3: if euphoniums should be written in bass clef, why aren't bass clarinets, tenor saxes, baritone saxes, etc.? And why do we not have only C trumpet parts, and teach young students that when they play a C in the staff they use the first valve on their Bb trumpet/cornet?
Those are my thoughts on the best clef for euphonium. Feel free to comment!
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Here in the US, the vast majority of performance opportunities are in wind bands, marching bands, and in doubling on either trombone or tuba. EXCEPT for brass band, those other opportunities are largely focused around bass clef and generally "think" concert pitch.
I also grew up playing baritone and cello, as well a smattering of piano from age 7. I became fluent in bass clef and tenor clef and performed euphonium, trombone, cello, and bassoon literature on my euphonium for 50 years. I became fluent in transposed treble when I picked up a trumpet during my senior year of university.
Unless there is certainty around playing in the brass band genre, either in the SA or outside of the US, I would encourage starting on bass clef, then becoming fluent in tenor and transposed treble.
Doug
This is the way the notation appears for a number of the first trombone parts for the community orchestra I'm in.
There is a reason.
I rather have her play Treble for wind related instruments. Better options as she grows older.