Are the treble and bass clef, as used in normal piano score, the only clefs around? Or are there others that have lost popularity, or are used in special cases, etc?
Here are the standard clefs available in the Sibelius music notation program. There are 22 of them, not counting guitar tablature. A few of the clef symbols are alternate ways of notating the same thing ("Treble down 8"), but as you can see, most are distinct. The other thing to note is that there are only three basic symbols for pitched instruments ("G", "F", and "C"), but the difference in clefs is where the symbol is oriented vertically on the clef and whether or not it has an octave transposition (up or down one or two octaves, designated by the numbers "8" or "15" above or below).
Then there is the special staff for percussion, which is rather the absence of a clef symbol since the percussion instruments being notated have no pitch.
Historically there used to be more clef symbols and positions than we use today. For example, as late as the early 1800s, the voice parts for soprano, alto, tenor and bass singers were notated each with their own unique clef symbol and position. But today we use the same clef for both soprano and alto, and the same clef for both tenor and bass (although alternately the tenor voice gets its own unique clef and staff, "Treble down 8", which all tenor singers much prefer).
These days we tend to simplify things and use the Treble and Bass for almost everything; the notable exception in the orchestra being the viola, which always gets its own "C" clef.
The following image shows the alto "C", bass "F", and treble "G" clefs alongside each other in the most common position. These are the most common clefs for pitched instruments. "Non-pitched percussion", (e.g., a drum set, but not xylophones or chimes), will often use the percussion clef (see Nick B.'s answer). (Images courtesy of Wikipedia.)
Historically, these clefs resembled the letters that remain their nicknames and the nicknames are not by accident.
C clef is always centered around middle C. Wherever the middle of the clef (between the upward and downward arcs) falls, there lies middle C. I have seen this fall on the center line, the second line from the top, the second space from the top (simulating treble clef written down an octave), and the second space from the bottom (simulating bass clef written up an octave).
F clef is always centered around the F below middle C. Whatever falls between the two dots is F. I have seen this on the second line from the top and the middle line.
G clef is always centered around the G above middle C. Whatever falls in the middle of the lower curl is G. I have seen this on the second line from the bottom and the bottom line.
The following image shows some of these other placements and the names associated with them.
In addition, you will sometimes see an ottava (octave) notation applied to the clef. The most common is a small 8 above or below the clef. If the 8 is above the clef, the intended pitch is an octave above that intended without the octave notation (8va); if the 8 is below the clef, the intended pitch is an octave below that intended without the octave notation (8vb).
When is ottava notation used? I have most frequently seen treble 8vb clef written for tenors in a mixed voice choir and men's barbershop music. Music for the soprano recorder is often written in treble clef, but it is more properly (and sometimes) written in treble 8va clef. Women's barbershop music often uses the bass 8va clef.
There are orchestral/wind ensemble instruments that regularly use the C clefs, in everyday situations. Some of the C clefs have conventional names for themselves, and certain instruments use them:
When the C clef is centered on the 3rd line of the staff it's called an alto clef, or viola clef.
Instruments that generally use this clef include:
Viola Sometimes other alto voice instruments will use this clef…
If a C clef is moved up to the 4th line of the staff, thus indicating that the C is written on the 4th line, it's then known as a tenor clef. When reading notes written in tenor clef it may be helpful to know that the notes are simply one line or space higher than they would be in treble clef but they sound an octave lower.
Instruments that generally use this clef include:
Bassoon Cello Trombones
There are three clefs in general use, the G clef, the F clef and the C clef.
The G clef is normally positioned on the second line up, indicating that this line is G. This is commonly called the "Treble clef". A common variation is the sub-octave treble clef, shown either with a small figure 8 below or as a repeated G clef symbol. This is used for tenor singers, and tells them to sing an octave lower than normal treble clef notation.
The F clef is normally positioned on the 4th line, when it is called the "Bass Clef".
The C clef is now only commonly found on the middle line "Alto clef" or 4th line "Tenor clef".
All these clefs have historically been used in other positions. You might still find the C clef positioned as "Baritone", "Soprano" or "Mezzo-soprano" clefs in older choral editions. The score-publishing program Sibelius offers these, also the "French Violin clef" (G clef on the bottom line) among its standard choices.
Treble and bass clefs each have a few variations. An 8 or 15 above the clef means 8va or 15ma respectively (one or two octaves higher). In bass clef, the 8 or 15 can be below the staff, telling the player to play one or two octaves lower.
Other clefs include soprano, mezzo-soprano, alto, tenor, and baritone.
Finally, there is also a percussion clef. There is normally a legend at the top of the sheet music that tells which lines on the staff correspond to each drum.
For more information about percussion clef, visit Wikipedia's Percussion Notation Page.
Besides the others shown here, there is another C clef, probably not used anymore, but found in "some nineteenth century arrangements of religious music for male chorus where the staff on which the two top voices are written is a C clef showing middle C on the second space from the top of the staff" (https://www.dolmetsch.com/musictheory14.htm). This C clef can also be considered a type of tenor clef, and is functionally equivalent to the "treble down 8" clef shown in this answer.
(Thanks to Daniel Caton for finding the Dolmetsch music theory page.)
The sheet music below, showing use of this clef, has a copyright date of 1969. Apologies for the quality of the scan.
Another example (in non-religious music for male chorus) can be found here.
They are the ones used on pianos and guitars, but some instruments have their range awkwardly situated for inclusion in one or the other - treble or bass. Thus, there is a C clef, which locates middle C on any line needed. As in the third line up, for example, where B in the treble lives normally. That then means the notes playable by the instrument mostly fit in the staff, without the use of never-ending leger lines, which are not so easy to read.Having said that, it's only one note from an octave C, so there seems little point. The notes could be read in a similar way to guitar notes, which are an octave out. It sort of works as a transposer in its own right.