A student asked about Islam's perspective on the Euthanasis of animals–is it permitted or prohibited? On what condition is it permitted if it is? I have never been asked the question before (they had some intesresting questions maa sha Allah), so I honestly could not even guide the student to answers.

Alhamdu-li-llah wa as-Salatu wa as-Salamu ‘ala Rasool-il-llah,
The kind treatment of animals is an Islamic injunction.
The Prophet PBUH said:
“عُذِّبَتْ امْرَأَةٌ فِي هِرَّةٍ سَجَنَتْهَا حَتَّى مَاتَتْ فَدَخَلَتْ فِيهَا النَّارَ لا هِيَ أَطْعَمَتْهَا وَسَقَتْهَا إِذْ حَبَسَتْهَا وَلا هِيَ تَرَكَتْهَا تَأْكُلُ مِنْ خَشَاشِ الأَرْضِ” رواه الشيخان عن أبي هريرة.
“A woman will be tormented in Hell because of a cat which she detained until she died. She did not feed it when she detained it, and she did not let it free to eat of the vermin of the earth.”
He PBUH also said:
” لا تَتَّخِذُوا شَيْئًا فِيهِ الرُّوحُ غَرَضًا” رواه مسلم عن ابن عباس.
“Do not use anything in which there is a soul as a target.” (Muslim).
Having made that clear, if an animal reaches old age, and becomes sick, it may be euthanized. That is to give it comfort and not burden anyone with its care. If we attempt to take care of all sick animals, we will not be able to care for the sick and dying children all over the world. That is insanity, not mercy. If we don’t distinguish between any living creatures and put the same value on a child and a puppy, why not save all other animals, including the mice and lizards and even insects?
Euthanasia is forbidden for humans, but not animals. However, in case the animal is to be euthanized, the killing must be swift and as merciful as possible. The Prophet  said:
“إِنَّ اللَّهَ كَتَبَ الْإِحْسَانَ على كل شَيْءٍ فإذا قَتَلْتُمْ فَأَحْسِنُوا الْقِتْلَةَ وإذا ذَبَحْتُمْ فَأَحْسِنُوا الذَّبْحَ وَلْيُحِدَّ أحدكم شَفْرَتَهُ فَلْيُرِحْ ذَبِيحَتَهُ.” رواه مسلم عن شداد بن أوس.
“If you kill, kill well, and if you slaughter, slaughter well. Let each one of you sharpen his blade and let him spare suffering to the animal he slaughters.”
Allah knows best.