I'm studying a string quartet by Haydn. This (obviously) doesn't have a basso continuo.
Why were basso continuos no longer used in the classical period? Was it something to do with the development of available instruments?
It seems that available instrumentists (more than instruments) were key as well as a figure of the composer as an individualistic and total master of the music he writes.
The development of cellists' virtuosity (such as Joseph Franz Weigl, friend of Haydn, Josef Fiala, Josef Reicha (friend of Mozart) and later the Duport brothers and Romberg) has been influential in this evolution and matched the mastery level and individuality of prominent violin and keyboard players/composers.
I see Haydn truly at the turning point of those forms when each part of the continuo (string bass, usually a cello, and keyboard) are not being lumped together anymore as a background harmony and music tapestry but as individual members of a group. This affects Haydn not only for chamber music but also his symphonies (the early ones have a continuo he used to play himself at performances, the latter ones had none).
As musicians both proficient at the violin and at the keyboard, Haydn or especially Mozart could not be satisfied to let an interpret apply traditional recipes at the keyboard or the cello to follow their composition.
Mozart by interaction and admiration of Haydn and later Beethoven renforced this tendency. This can be observed in most of Mozart's chamber music, notably trios with piano.
Composers discovered that writing parts corresponding to traditional continuo instruments was an important mean of expression for their ideas and allowed a more distinctive style and experimentations.
By the way, several historians insist on the trio sonata origin of the quartet form (dumping the keyboard because of circumstances).
In this sense a string quartet follows from this trend, as it is really an explicit form written out by the composer in four voices with only bowed string instruments. Haydn established the genre and Mozart followed studiously.
Perhaps you should compare
In the two cases you have 2 to 4 string instruments and a keyboard. But the first group has explicit parts for each player. You can argue that in the first group there you can watch composers such as Mozart progressively developing a Mozartian style for writing cello or viola parts. Mozart practiced viola and noted that he liked to play the viola when playing his chamber music with friends to be in the center of the harmony.
An interesting fact that I learned making some verifications for this answer: Johann André, music publisher and purchaser of most of Mozart's manuscript after his death transcribed three Mozart trio with piano into string quartets, this is his opus 32. It has been recorded in 2008 by the Quartetto Tomasini for the Hungaroton Label.
It's just a change in style, like the way blues and rock bands changed from having guitars only in the background to having guitars up front.
The basso continuo was a standardized sort of accompaniment, typically given a bass line and chord symbols only and filing in the chords ad libitum. J.S. Bach was already beginning to exert more control, writing keyboard parts out instead of leaving it up to the performer to improvise. Some of his solo sonatas are written for solo and keyboard only, with no separate bass or cello part.
Even for those Bach pieces you could add a continuo and it would still work. Mozart and Haydn moved away from that style of music toward a style where all the lines -- even the cello part -- are foreground voices without a background. Listen to a Haydn string quartet and try playing chords along with it: it doesn't work quite as well as it did with Bach.