Can I make dua during sujood?

Yes, and it's one of the best moments to do it. The Prophet ﷺ taught that sujood (prostration) is when you are closest to Allah and told us to make a lot of dua in it. After saying the normal "Subhana Rabbiyal A'la", you can add your own requests. Scholars differ a little on what to say in obligatory prayers, so many people keep longer personal duas for voluntary prayers.

Why sujood is special

Putting your forehead on the ground is the most humble position a person can take, and that's exactly when Allah is nearest. The Prophet ﷺ said: "The nearest a servant comes to his Lord is when he is prostrating himself, so make supplication (in this state)" (Sahih Muslim 482). In another hadith he said: "while prostrating yourselves be earnest in supplication, for it is fitting that your supplications should be answered" (Sahih Muslim 479). See Can I make dua for worldly things like a job or a car?

So sujood isn't only a ritual step. It's a private conversation, built into every prayer you pray. See What if I don't feel anything when I pray?

What to say

First, the basic dhikr: "Subhana Rabbiyal A'la" (Glory be to my Lord, the Most High), usually three times.
A short prophetic dua: the Prophet ﷺ often said in bowing and prostration "Subhanaka Allahumma Rabbana wa bihamdika, Allahumma-ghfir li", meaning "Exalted are You, O Allah our Lord, and by Your praise. O Allah! Forgive me" (Sahih al-Bukhari 817).
Asking for forgiveness: he also used to say in sujood: "O Lord, forgive me all my sins, small and great, first and last, open and secret" (Sahih Muslim 483).
Your own needs: guidance, health, a job, your family, a hard decision. Keep it sincere and specific.

Obligatory vs. voluntary prayers

All four schools agree dua in sujood is good, but they differ on the details:

Hanafi school: in the obligatory (fard) prayers, stick to the set dhikr, and save personal duas for voluntary prayers like the sunnah prayers or tahajjud.
Maliki, Shafi'i and Hanbali schools: dua in sujood is fine in both obligatory and voluntary prayers, as long as it doesn't drag out a congregational prayer.

Whether a non-Arabic dua is allowed inside the prayer is a separate question.

Can I use Quran verses as dua in sujood?

The Prophet ﷺ said he was "forbidden to recite the Qur'an in the state of bowing and prostration" (Sahih Muslim 479). Many scholars still allow the duas found in the Quran, like "Rabbana atina fid-dunya hasanah...", if you say them as a dua and not as recitation. Others prefer duas from the hadith to be on the safe side.

ٱدْعُوا۟ رَبَّكُمْ تَضَرُّعًۭا وَخُفْيَةً ۚ إِنَّهُۥ لَا يُحِبُّ ٱلْمُعْتَدِينَ "Call upon your Lord in humility and privately; indeed, He does not like transgressors." Quran 7:55

When you're praying behind an imam

In congregation, follow the imam. Say your dhikr and a short dua, but don't stay down after he rises. If you want long, unhurried sujood, the night prayer or a voluntary prayer at home is perfect for it.

Key points

Sujood is when you're closest to Allah.
Say the basic dhikr first, then add your dua.
Hanafis keep personal duas to voluntary prayers.
Follow the imam in congregation.

Tip: Pick one short dua to memorize this week and say it in the sujood of your sunnah prayers. Once it feels natural, add another one.