From my own experience, originating from Mahayana understanding, I have come to a conclusion that cultivating compassion itself is kind of a folly. A waste of time.
I am not saying compassion is a mistake, but especially from Mahayana PoV where compassion is seen as a "virtuous mind state" that IS the way to liberation and point of existence (leading to bodhicitta) it seems illogical.
The Buddha explained that compassion is an antidote to ill will, nothing more. He did not explain that compassion is a step towards liberation, he actually states that it leads to a more fortunate rebirth but not specifically to liberation, hence why it is not a part of the eightfold path. Wiki page here of the 4 immeasurables.
This is not to say that compassion is an unwholesome mindstate, it is not a klesha in the conventional terms, a defilement the ruins stillness and peace of mind. If anything compassion bring pleasure and joy to the person experiencing it.
Still my experience with compassion, as well as the logic of anatta has led me to believe it is a waste of time to put effort into cultivating, unless of course you are a hateful person, or have hate for a specific person, again it can be used as an antidote to ill will.
In fact compassion arises naturally anyway while leading a virtuous life.
One only needs to think of another or our own suffering in the past to experience the mindstate naturally.
As for a more conventional look at compassion, in society I have experienced that compassion is often seen as a weakness. This will depend on each of our own personal social circumstances but for my own PoV being a man and living in a lower class life in a western society, compassion is not often seen as a virtue to others. This will obviously differ depending on other circumstances. Say a female in a nursing profession having compassion for a patient will be seen as virtuous, as an example.
Lastly you can keep in mind that you can have compassion for others but actually not show others that you have compassion for them. Compassion is an inner mental factor, it is not an external action, or word. There is often no need to show others compassion and in some cases showing compassion can harm others as they feel you pity them. It is actually a difficult mindstate to express emotionally or even with words.
Ultimately you can perfect compassion and have what is known as universal (great) compassion by the mahayana yet nobody other than you will ever know, as there is no way to show this "attainment" to others.