First of all, a question arises:
How or why is the name or concept of Leela relevant in Hinduism?
It's because Brahman is defined as fully omniscient and omnipotent by many Vedic verses. Brahman is defined as being beyond mind and hence to be understood by Neti Neti or Not-this Not-this. So for this powerful and omniscient Brahman there cannot be any purpose to create this world (as Brahman is all powerful at the first place). So, the creation of this world by Brahman is defined as just being playful sport or Leela of Brahman. Just as we playfully play with children without any purpose similar is the case.
Is there a scriptural definition of this word līlā?
The word Leela is explained itself in the Brahma Sutra 2.1.33 which states:
लोकवत्तु लीलाकैवल्यम् ।
Lokavattu lilakaivalyam II.1.33
But (Brahman's creative activity) is mere sport, such as is seen in the world (or ordinary life).
Here,
Lokavat: as in the world, as in ordinary life;
Tu: but;
Lilakaivalyam: mere pastime.
(Lila: sport, play; Kaivalyam: merely; Lilakaivalyam: mere pastime.)
Here is Sri Adi Shankara commentary on this Sutra:
The word 'but' discards the objection raised.--We see in every-day life that certain doings of princes or other men of high position who have no unfulfilled desires left have no
reference to any extraneous purpose; but proceed from mere sportfulness, as, for instance, their recreations in places of amusement. We further see that the process of inhalation and exhalation is going on without reference to any extraneous purpose, merely following the law of its own nature. Analogously, the activity of the Lord also may be supposed to be mere sport, proceeding from his own nature 1, without reference to any purpose. For on the ground neither of reason nor of Scripture can we construe any other purpose of the Lord. Nor can his nature be questioned 2--Although the creation of this world appears to us a weighty and difficult undertaking, it is mere play to the Lord, whose power is unlimited. And if in ordinary life we might possibly, by close scrutiny, detect some subtle motive, even for sportful action, we cannot do so with regard to the actions of the Lord, all whose wishes are fulfilled, as Scripture says.--Nor can it be said that he either does not act or acts like a senseless person; for Scripture affirms the fact of the creation on the one hand, and the Lord's omniscience on the other hand. And, finally, we must remember that the scriptural doctrine of creation does not refer to the highest reality; it refers to the apparent world only, which is characterised by name and form, the figments of Nescience, and it, moreover, aims at intimating that Brahman is the Self of everything.
Here is Sri Ramanujacharya commentary on the same:
The motive which prompts Brahman--all whose wishes are fulfilled and who is perfect in himself--to the creation of a world comprising all kinds of sentient and non-sentient beings dependent on his volition, is nothing else but sport, play. We see in ordinary life how some great king, ruling this earth with its seven dvîpas, and possessing perfect strength, valour, and so on, has a game at balls, or the like, from no other motive than to amuse himself; hence there is no objection to the view that sport only is the motive prompting Brahman to the creation, sustentation, and destruction of this world which is easily fashioned by his mere will.
Swami Sivananda commentary on the Sutra as:
The objection raised in Sutra 32 is replied to. The word ‘tu’ (but) removes the above obejction. Brahman has created the world not out of any desire or motive. It is simply His pastime, proceeding from His own nature, which is inherent in and inseparable from Him, as it is seen also in the world that some times a rich man or a prince, does some action with out any motive or pur pose, simply out of a sportive impulse. Just as children play out of mere fun, or just as men breathe without any motive or purpose, because it is their very nature, just as a man full of cheerfulness when awakening from sound sleep, begins to dance about with out any objective, but from mere exuberance of spirit, so also Brahman engages Himself in creating this world not out of any purpose or motive, but out of sporting or Lila or play proceeding from His own nature. Although the creation of this universe appears to us a weighty and difficult undertaking, it is mere play to the Lord, whose power is infinite or limitless. If in ordinary life we may possibly by close scrutiny detect some subtle motive even for sportful action (playing at a game of balls is not altogether motive less, be cause the prince gets some pleasure by the play), we can not do so with regard to the ac tions of the Lord. The scripture declares that all wishes are ful filled in the Lord and that He is all-full, self-contained and self-sufficient. It should not be forgotten how ever that there is no creation from the standpoint of the Absolute, because name and form are due to Avidya or ignorance and because Brahman and Atman are really one. The opponent again raises an objection. The theory that Brahman is the creator is open to the objection that He is either partial or cruel, because some men enjoy happiness and others suffer misery. Hence this theory is not a congruous one. This objection is removed by the following Sutra.
So, the act of Brahman is defined as his mere play or sport.
Are there any examples from scriptures where certain actions are classified as līlās?
As seen above from Vedanta Sutra we know that activities of Brahman are termed as Leela. Now, scriptures such as Vishnu Purana, Bhagvata Purana etc., identify Brahman as Vishnu. So they explain activity of Vishnu as his mere leela.
Similarly, scriptures such as Shiva Purana, Linga Purana etc., identify Brahman as Shiva. So, they explain activity of Shiva as his mere leela. Similarly, Devi Bhagvatam and Saptasati identify Brahman as Devi. So, they explain activity of Devi as mere leela.
For instance, here I'm giving an example from Srimad Bhagvatam. In the beginning part of Srimad Bhagvatam, Sage Suta Goswami is asked to describe leela of Lord:
अथाख्याहि हरेर् धीमन्न् अवतार-कथाः शुभाः।
लीला विदधतः स्वैरम् ईश्वरस्यात्म-मायया।। (S.B. 1.1.18)
O wise Sūta, please narrate to us the transcendental pastimes of the Supreme Godhead’s multi-incarnations. Such auspicious adventures and pastimes of the Lord, the supreme controller, are performed by His internal powers.
In Bhagvatam 1.2.32 it is stated taking incarnation by Lord are just his Leela:
भावयत्य् एष सत्त्वेन लोकान् वै लोक-भावनः।
लीलावतारानुरतो देव-तिर्यङ्-नरादिषु।। (S.B. 1.2.32)
Thus the Lord of the universes maintains all planets inhabited by demigods, men and lower animals. Assuming the roles of incarnations, He performs pastimes to reclaim those in the mode of pure goodness.
Similarly, in the ending part of Srimad Bhagvatam it is stated every action by Lord Vishnu described in this Purana are his Leelas:
आदि-मध्यावसानेषु वैराग्याख्यान-संयुतम्।
हरि-लीला- कथा-व्राता-मृतानन्दित-सत्-सुरम् ।। (S.B. 12.13.11)
From beginning to end, the Śrīmad-Bhāgavatam is full of narrations that encourage renunciation of material life, as well as nectarean accounts of Lord Hari’s transcendental pastimes, which give ecstasy to the saintly devotees and devas.
If an individual performs a certain action which some people view as dharma and others view the same action as adharma, do scriptures allow the person to call his action leela?
As already clear from above commentaries and Sutra itself, the action of people differs from action of Brahman i.e., Action of Brahman is purposeless or without any motive, however, a person has some motive or purpose. Even if a person does Niskama Karma he may have motive or he may be motivated to do Niskama Karma. But Advaitically speaking if a person does some activity after fully identifying himself as Brahman (I am Brahman), then his activity would be also like that of play or sport as he wouldn't have any purpose to do anything. In such case we can say that person's activity as Leela.
