Short answer
The Quran teaches that some of the flames seen streaking across the sky are missiles that drive away devils trying to eavesdrop on the angels. Muslims believe this because the Quran says it; it does not deny that meteors are also physical objects. Making a wish on a shooting star has no basis in Islam.
Muslims believe what the Quran and sahih hadith say about the shihab (burning flame) striking eavesdropping devils. Believing that stars or meteors bring luck, grant wishes or cause events on earth is superstition, which the Sunnah rejects.
The four schools agree on this.
Explanation
Allah says: "We have certainly beautified the nearest heaven with stars and have made [from] them what is thrown at the devils" (67:5). The jinn themselves describe it in Surah al-Jinn: they used to sit in places to listen, "but whoever listens now will find a burning flame lying in wait for him" (72:9). Surah al-Saffat adds that the one who snatches a word is pursued by a piercing flame (37:10).
The Prophet ﷺ explained it. When Allah decrees a matter in heaven, the angels beat their wings in submission, and they speak of it among themselves. The eavesdroppers among the jinn try to catch a word and pass it down. Sometimes a flame catches one before he passes it on, and sometimes he passes it to a fortune-teller, who then mixes it with a hundred lies. That is why fortune-tellers are sometimes right, and why trusting them is forbidden.
Does this deny the science of meteors? No. Islam does not ask you to choose between a physical cause and an unseen purpose. The same streak of light can be a rock burning in the atmosphere and also a means Allah uses to guard the heavens. The early scholar Qatadah said, in a statement recorded by al-Bukhari, that the stars were created as an adornment of the sky, as missiles against devils and as signs for navigation.
Wishing on a shooting star, or reading omens from it, is a superstition. Ask Allah directly instead. See are superstitions haram?, astrology in Islam and the universe in Islam.
The evidence
Quran, Surah Al-Mulk 67:5
وَلَقَدْ زَيَّنَّا ٱلسَّمَآءَ ٱلدُّنْيَا بِمَصَٰبِيحَ وَجَعَلْنَٰهَا رُجُومًۭا لِّلشَّيَٰطِينِ ۖ وَأَعْتَدْنَا لَهُمْ عَذَابَ ٱلسَّعِيرِ ﴿٥﴾
And We have certainly beautified the nearest heaven with stars and have made [from] them what is thrown at the devils and have prepared for them the punishment of the Blaze.
States that some stars serve as missiles against devils.
Quran, Surah Al-Jinn 72:9
وَأَنَّا كُنَّا نَقْعُدُ مِنْهَا مَقَٰعِدَ لِلسَّمْعِ ۖ فَمَن يَسْتَمِعِ ٱلْءَانَ يَجِدْ لَهُۥ شِهَابًۭا رَّصَدًۭا ﴿٩﴾
And we used to sit therein in positions for hearing, but whoever listens now will find a burning flame lying in wait for him.
The jinn describe being driven off by flames.
Sahih al-Bukhari 4701
Sahih... إِذَا قَضَى اللَّهُ الأَمْرَ فِي السَّمَاءِ ضَرَبَتِ الْمَلاَئِكَةُ بِأَجْنِحَتِهَا ...
"When Allah decrees a matter in heaven, the angels beat their wings in submission to His word."
Opens the hadith that explains the eavesdropping and the flames.
Where the ruling is recorded
- Tafsir Ibn Kathir, Commentary on Surah al-Mulk, verse 5. Explains the stars as adornment and as missiles against the devils.
- Tafsir al-Qurtubi, Commentary on Surah al-Jinn. Discusses the guarding of the heavens and the flames that strike the eavesdroppers.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.