Can I save a spot in the mosque?

Only in a limited way. If you were sitting in a spot and step out briefly, for wudu or the bathroom, you have more right to it when you come back. But reserving places in advance with bags or prayer mats, or asking someone to move for you, goes against the mosque's etiquette.

The mosque belongs to everyone

In the mosque, places go to whoever arrives first. Nobody owns a row, not even regulars or elders. The Prophet ﷺ said a man should not make another man get up from his place and then sit there himself, but people should make room and spread out (Sahih al-Bukhari 6270). The Quran teaches the same generous spirit:

يَـٰٓأَيُّهَا ٱلَّذِينَ ءَامَنُوٓا۟ إِذَا قِيلَ لَكُمْ تَفَسَّحُوا۟ فِى ٱلْمَجَـٰلِسِ فَٱفْسَحُوا۟ يَفْسَحِ ٱللَّهُ لَكُمْ "O you who have believed, when you are told, 'Space yourselves' in assemblies, then make space; Allah will make space for you." Quran 58:11

When you can keep your spot

If you were already sitting somewhere and leave for a short, real need, you may come back to the same place. The Prophet ﷺ said that whoever gets up from his seat and then returns to it has more right to it (Sahih Muslim 2179). Leaving a light item like a book or a jacket to show you'll be right back is generally accepted for this short absence. Many scholars do not allow this for long breaks or for "booking" a place before you even arrive.

What to avoid

Laying out mats or bags early to reserve a front spot, then arriving late. Many scholars call this disliked, and some say the latecomer can't claim it.
Saving a whole row for your friends or family while others stand at the back.
Stepping over people to reach a "reserved" place. When a man did this during the khutbah, the Prophet ﷺ told him to sit down because he was causing harm (Sunan Abi Dawud 1118).
Walking in front of someone praying to get to your spot. See Is it haram to walk in front of someone praying?

Want a front-row place? Come early

The front rows carry a special reward. The Prophet ﷺ said that if people knew the reward of the first row, they would draw lots for it (Sahih al-Bukhari 615). The way to earn it is by arriving early, not by holding it. Early arrival also gives you time for two rak'ahs on entering. See What is tahiyyat al-masjid? On Fridays this is part of the day's sunnah. See What are the sunnahs of Friday?

Special cases

Mosques may set rules for safety or access, such as chairs for elderly worshippers, a space near the door for people with disabilities, or volunteers who manage overflow rooms on busy Fridays. Following those rules is fine and is not the same as personal reserving.

Key points

Places in the mosque go to whoever arrives first.
A short break for wudu doesn't lose you your spot.
Don't reserve places in advance or move people.
For the front row, come early instead.

Tip: If someone is in "your usual" spot, smile and sit elsewhere. If you see a gap near you when a latecomer arrives, shuffle over. Small acts like these make a mosque feel welcoming.