None. Ancient Greeks subdivided the world into and the superlunar the sublunar spheres, the heavens and the Earth. The former a place of perfection and order, the latter, not so much. Since meteors and comets did not display the reliable regularity of the fixed stars, or even of the "wandering" planets they clearly did not belong to the heavens. Greeks therefore saw them as the (upper) atmospheric phenomena, where air and fire had their "natural place", along with thunder and lightning, clouds, aurora, and rainbows. "Meteorology" and "meteors" got their names from Greek "meteoros", raised up high.
Aristotle's view prevailed in both Europe and the Islamic East throughout the middle ages. Comets acquired their celestial status first after Halley convincingly argued that the comet seen in 1705 (and now named after him) was the same comet seen in 1531, 1607 and 1682, and predicted that it will be seen again in 1758. Displaying such reliable periodicity quickly moved the comets to the realm of astronomy.
It took longer for meteors, Kepler still believed that they were atmospheric phenomena, and Newton hypothesized that heavens were filled with stars, planets, and comets only. However, already in 1676 Wallis wrote sceptically:"that which makes it to me the more surprising is this; that I find the same [meteor] to have been seen in most parts of England". And in 1714 Halley wrote three papers, where he reasoned that atmosphere barely extended as high (45m/70km) as most of the "fiery meteors" appeared. But.. history is rarely linear, in 1719 Halley changed his mind and went back to Aristotle, and it took the good part of 18th century to finally move meteors into astronomy. See Beech's Makings of Meteor Astronomy.
Here is the opening of Aristotle's Meteorology (boldface mine):
We have already discussed the first causes of nature, and all natural motion, also the stars ordered in the motion of the heavens, and the physical element-enumerating and specifying them and showing how they change into one another-and becoming and perishing in general. There remains for consideration a part of this inquiry which all our predecessors called meteorology. It is concerned with events that are natural, though their order is less perfect than that of the first of the elements of bodies. They take place in the region nearest to the motion of the stars. Such are the milky way, and comets, and the movements of meteors.
And here is his theory of the shooting stars:
"We must think of what we just called fire as being spread round the terrestrial sphere on the outside like a kind of fuel, so that a little motion often makes it burst into flame just as smoke does: for flame is the ebullition of a dry exhalation. So whenever the circular motion stirs this stuff up in any way, it catches fire at the point at which it is most inflammable. The result differs according to the disposition and quantity of the combustible material... if the whole length of the exhalation is scattered in small parts and in many directions and in breadth and depth alike, we get what are called shooting-stars... Shooting-stars further suffix to prove that the celestial sphere is not hot or fiery: for they do not occur in that upper region but below: yet the more and the faster a thing moves, the more apt it is to take fire."