Is there any benefit in abstaining from listening songs?

Correcting earlier... “Could you figuratively use a feather of awareness to the music compared, e.g., a hammer, that is, a lighter note.”

If everything passes - in time - then if one were to let it it pass, (feather) compared to paying it so much attention (hammer) maybe the mental activity could be noted of the object (thoughts, music, etc.) but not necessarily sustained, maintained, retained.

You can listen to music while learning dharma thus:

Sound - ear - interpretation - feeling - craving or aversion.

How so can one learn from this?

The source of wanting or averting is in the feeling (vedana) and since nothing can be sustained to our liking, then it is all impersonal or not-self. From a more direct perspective, it is impermanent, suffering and therefore not who you 'think' you are. The same is true of all the other senses.

You could really go some here by seeing how this process fits into the 12 nidanas.

"Nidanas" are co-dependent events or phenomena, which act as links on a chain, conditioning and depending on each other. When certain conditions are present, they give rise to subsequent conditions, which in turn give rise to other conditions. Wikipedia

Additionally, in the Assutavā Sutta the Buddha said...

When this exists, that comes to be. With the arising of this, that arises. When this does not exist, that does not come to be. With the cessation of this, that ceases.

The answer to your question is yes there is benefit from not listening but also there is an opportunity to learn from your physiology when listening.

Going forth (i.e., becoming a monk)

They avoid dancing, singing, music, and seeing shows --MN27

For a lay person this may sound like quite the burden. However, it's not. Music is by its nature compelling. Wanting to listen to music is wanting to be caught up in it, to be emotionally tied and bound. Even professional musicians understand this and need to develop a necessary measure of restraint, distance and awareness to advance their skills. Similarly, listeners naturally develop a certain distance when listening mindfully. "Oh this song wants me to feel seduced by romance". Etc.

Note that the sutta uses "avoid". As in "avoid stepping in bear traps with bare feet." There is no sin or virtue here. I also avoid firecrackers because they can cause injury. Music isn't bad or good. In fact, one can actually listen to music mindfully to understand where hidden delights are bubbling. Then one can go meditate on those delights and relinquish them if possessive about them.

Is there any reason to keep on nourishing of what is just of short pleasure, does not lead to liberation?

Is there any reason to nourish something being an obstacle on the path? Having told by the wise and particularly seen for oneself?

By not nourishing, not giving note, not giving into, withdraw, cutting of any feed, relations break apart, good as well as bad.

Now think how dhamma, suttas, good reminders would pup up in the head, if starting to seek for relation to it? So it's good to seek for relation where they repeat, enjoy and remember each other of what is worthy to nourish, remember and store, to face the unconditioned possible for yourself and be not without good entertaining /lifelihood for the mind.

Yes, listening to music is called entertainment or sense gratification. You need to understand the purpose of listening to music. Is it helping to grow spiritually? Music will create emotions in mind, leading to suffering. There are no cheats or listening to music once in a while. You have to be serious for practicing dhamma. Ancient Buddhist monks also practiced, avoiding to watch beautiful or colourful things for controlling mind. Reading books is called intellectual entertainment. Dhamma is for disciplined candidate. Discipline will liberate your mind. There has to be a fire of discipline to grow spiritually.

Only a personal experience, but for the substantial period that music was not a part of this life a few things happened. The mind was more clear and focused. Meditation came easier and deeper. There was one less distraction and attachment to deal with.

It is my practice that not listening to music is in strict accordance with the precepts. With this being said, it was only a challenge at the beginning stage and includes the position you describe.

All the best to you friend.

If you stop listening to songs, eventualy the odds of fragments of song poping in the mind would shrink, but is that realy what you are looking for ?

A better way to approach this would be to let go of the aversion you feel toward having songs beeing replayed in your mind.

When it occurs, do as you do with any other distraction or thought. Watch is appear and disappear by itself, watch for any sensation or thought that may arise because of it.

Even if you decide not to listen to music anymore, you will be exposed to it against your will from time to time, so you can't really count on "dropping the listening habit" strategy alone.

Please also consider that one can listen to music in order to feel sensual pleasure, which is unskillful, but also to engage with a particular art form which won't prevent your spiritual growth at all. See this question Buddhist view on Art and being an Artist.

Music can increase one's motivation in his spiritual developement, by giving access to particular emotions one would have never felt otherwise.

Just a few examples on the top of my head : If you are on a bad day, listening to a particular song could uplift your mind and give you the motivation to sit and meditate. If you have trouble feeling goodwill towards other human beeings, try listening to Beethoven's 9th symphony. Also 5th symfony can easely be interpreted by the story of a person who gradually overcomes his bad kamma. 6th symphony is about joy associated with nature. And so on.

As with any other actions, you should contemplate the effects of listening to music. With time you will build up discernement, and avoid situations when listening to music is unskillful. Be good to yourself and don't beat yourself over it unnecessarily. When you see clearly the drawbacks of music, things will resolve quite naturally.