There are many incidents that may actually indicate Rama's divine characteristics and I am going to list them down with references from the Valmiki Ramayan.
FIRST incident is from the Bala Kanda Sarga 49 when Ahalya is freed from her curse. The verse mentions a shower of flowers from the sky when Ahalya washes Rama's feet:
पुष्प वृष्टिः महती आसीत् देव दुंदुभि निस्वनैः |
गन्धर्व अप्सरसाम् च एव महान् आसीत् समुत्सवः || १-४९-१९
There chanced an abundant floral fall form firmament to the drumbeats of god's drums, and the celestials like gandharva-s, apsara-s revelled in a splendid festivity that is superb. [1-49-19]
In Sarga 51 we hear Shatananda the chief-priest of Janak and son of Gautam and Ahalya asking Vishwamitra this question:
अपि मे गुरुणा रामः पूजितः कुशिकात्मज |
इह आगतो महातेजाः पूजाम् प्राप्य महात्मनः || १-५१-८
"Oh, Kaushika, my father came to my mother's place from Himalayas! Has the great resplendent father of mine worshipped Rama because the redemption of my mother is per the kindness of Rama! Has this great-souled Rama reverenced that great resplendent father of mine by according a redemption, ahalyaa daana, the endowment of Ahalya to her husband. [1-51-8]
He then addresses Rama thus:
स्वागतम् ते नरश्रेष्ठ दिष्ट्या प्राप्तो असि राघव |
विश्वामित्रम् पुरस्कृत्य महर्षिम् अपराजितम् || १-५१-१३
"Hail to thee! Oh, best one among men Rama, your arrival is a godsend, not only to Mithila but to entire humanity, and oh, Raghava, as an undefeatable great-sage Vishvamitra spearheads you, so shall your mission be undefeatable, thus hail to thee! [1-51-13]
The SECOND incident is of Rama lifting & stringing the Shiva Dhanush that even the gods couldn't lift. As detailed in Sarga 67:
न एतत् सुर गणाः सर्वे स असुरा न च राक्षसाः |
गंधर्व यक्ष प्रवराः स किन्नर महोरगाः || १-६७-९
क्व गतिः मानुषाणाम् च धनुषो अस्य प्रपूरणे |
आरोपणे समायोगे वेपने तोलने अपि वा || १-६७-१०
"None in all the assemblages of gods, inclusive of demigods, demons, gandharva-s, yaksha-s, kinnaraa-s, or reptilian demigods, is capable enough to take aim with this bow and all are rendered incapable... and then, in brandishing this bow for a proper grip, or in bracing its bowstring to the other end, or in twitching the bowstring for its tautness, or in placing the arrow on bowstring at a proper place for a proper stretchability, or even in taking a good aim with it... what will be that capability of one from among humans... [1-1-9, 10]
बाढम् इति एव तम् राजा मुनिः च समभाषत |
लीलया स धनुर् मध्ये जग्राह वचनात् मुनेः || १-६७-१५
"All Right!" said the saint and king to Rama in chorus, and Rama upon the word of the sage grasping it at the middle handgrip playfully grabbed the bow. [1-67-15]
पश्यताम् नृ सहस्राणाम् बहूनाम् रघुनंदनः |
आरोपयत् स धर्मात्मा स लीलम् इव तत् धनुः || १-६७-१६
आरोपयित्वा मौर्वीम् च पूरयामास वीर्यवान् |
तत् बभंज धनुर् मध्ये नरश्रेष्ठो महायशाः || १-६७-१७
While many thousands of men are witnessing that right-minded Rama the legatee of Raghu stringed the bow effortlessly. [1-67-16] Further, that dextrous one has stringed that bow with bowstring and started to stretch it up to his ear to examine its tautness, but that glorious one who is foremost among men, Rama, broke that bow medially. [1-67-17]
The THIRD incident is of him using Lord Vishnu's bow to take away the powers of Lord Parshu-Raam, who is himself an Avatar, that clearly show his divine powers. As per Sarga 75
श्रुत्वा तु धनुषो भेदम् ततो अहम् द्रुतम् आगतः || १-७५-२६
तत् एवम् वैष्णवम् राम पितृ पैतामहम् महत् |
क्षत्र धर्मम् पुरस् कृत्य गृह्णीष्व धनुर् उत्तमम् || १-७५-२७
"On hearing about the breakage of Shiva's longbow, then I promptly came here. Thereby, oh, Rama, wield this supernatural and superlative longbow of Vishnu, which is passed on to me from my forefathers and my father. Keep your fealty to Kshatriya-hood in view, and wield this as you have wielded Shiva's longbow... [1-75-26b, 27]
Rama then takes the long bow of Vishnu with the following words:
वीर्य हीनम् इव अशक्तम् क्षत्र धर्मेण भार्गव |
अवजानासि मे तेजः पश्य मे अद्य पराक्रमम् || १-७६-३
इति उक्त्वा राघवः क्रुद्धो भार्गवस्य वर आयुधम् |
शरम् च प्रतिजग्राह हस्तात् लघु पराक्रमः || १-७६-४
Oh, Bhaargava, demeaning me as though I am timorous, hence incapable to handle the bow, and hence I am an ignoble one for Kshatriya-hood, is meaningless... come on, now you may see my spiritedness and sprightliness" So said Rama to Bhaargava. [1-76-3] Raghava, the nimble-handed vanquisher of his opponents, speaking thus in high dudgeon expropriated that estimable weapon, namely the longbow of Vishnu, from the hand of Bhaargava Rama, along with the long-arrow that is already fitted on it... [1-76-4]
Even the gods themselves appear to witness this momentous occasion of Rama wielding Vishnu's bow to threaten one of his incarnations!
वर आयुध धरम् रामम् द्रष्टुम् स ऋषि गणाः सुराः |
पितामहम् पुरस्कृत्य समेताः तत्र सर्वशः || १-७६-९
गंधर्व अप्सरसः चैव सिद्ध चारण किन्नराः |
यक्ष राक्षस नागाः च तत् द्रष्टुम् महत् अद्भुतम् || १-७६-१०
Gods together with the assemblages of sages have come keeping the Grandparent Brahma at their fore, likewise the gandharva-s, apsara-s, siddha-s, caarana-s, kinnaraa-s, yaksha-s, sprites and reptilian beings have also come to see Rama who is now wielding the extraordinary longbow of Vishnu, and extremely amazing event that is going to ensue. [1-76-9, 10]
Even Parshurama realizes Rama's divinity and mentions thus:
अक्षय्यम् मधु हन्तारम् जानामि त्वाम् सुरेश्वरम् |
धनुषो अस्य परामर्शात् स्वस्ति ते अस्तु परंतप || १-७६-१७
एते सुर गणाः सर्वे निरीक्षन्ते समागताः |
त्वाम् अप्रतिम कर्माणम् अप्रतिद्वन्द्वम् आहवे || १-७६-१८
न च इयम् तव काकुत्स्थ व्रीडा भवितुम् अर्हति |
त्वया त्रैलोक्य नाथेन यत् अहम् विमुखी कृतः || १-७६-१९
"I have realized your touch of nature as that of the Immutable Supreme Being, God of Gods, the Exterminator of the demon Madhu, namely Vishnu, by the touch of your handling that bow. oh, enemy-inflamer, blessedness alone betides you... [1-76-17] "All of these gods who have come collectively are beholding you and your next move, for you are an unequalled one in you achievements and to whom there is no counter-dueller in conflicts... [1-76-18] "Oh, Kakutstha Rama, you are the lord of the triad of worlds, such as you are, you faced me down, and it is malapropos to say that this is a disgrace to me... [1-76-19]
FOURTH example is of him killing the giant Viradh in Sarga 4 of the Aranya Kand. The demon who was actually a Gandharva named Tumburu then addresses Rama saying:
अभि शापाद् अहम् घोरम् प्रविष्टो राक्ष्सीम् तनुम् |
तुंबुरुः नाम गन्धर्वः शप्तो वैश्रवणेन हि || ४-३-१६ प्रसाद्यमानः च मया सोऽब्रवीत् माम् महायशाः |
यदा दाशरथी रमः त्वाम् वधिष्यति संयुगे || ३-४-१७
तदा प्रकृतिम् आपन्नो भवान् स्वर्गम् गमिष्यति |
"By curse I had to enter this ghoulish demon's body, but I am a celestial gandharva, named Tumburu and Kubera cursed me... [3-4-16] "When entreated by me, he that greatly renowned Kubera said to me, 'When Dasharatha's Rama kills you in a fight... then you will attain your nature of celestial body and you will go to heavens' [3-4-17, 18a]
FIFTH in the Aranya Kand again Rama destroys fourteen thousand demons in the army of Khar and Dushan all by himself while the gods watch from above. As per Sarga 25 :
महाकपालस्य शिरः चिच्छ्हेद रघुनंदनः || ३-२६-२०
असंख्येयैः तु बाण ओघैः प्रममाथ प्रमाथिनम् |
स्थूलाक्षस्य अक्षिणी स्थूले पूरयामास सायकैः || ३-२६-२१
स पपात हतो भूमौ विटपी इव महाद्रुमः |
दूषणस्य अनुगान् पंच सहस्रान् कुपितः क्षणात् || ३-२६-२२
हत्वा तु पंच सहस्रान् अनयत् यम सदनम् |
Rama, the legatee of Raghu, chopped off the head of Mahakapaala, with incalculable spates of arrows he eliminated Pramaathi, and filled the bulging eyes of Sthulaaksha with arrows by which Sthulaaksha is killed and fell down onto ground like a burly tree with its broad branches. Then still infuriated Rama forthwith eliminated the five thousand hench-demons of Duushana with five thousand arrows, and thus he led them to the residence of Time-god, namely the hell. [3-26-20b, 21, 22, 23a]
रक्षसाम् तु शतम् रामः शतेन एकेन कर्णिना |
सहस्रम् तु सहस्रेण जघान रण मूर्धनि || ३-२६-३१
With a hundred rudder like arrows Rama hit down a hundred demons, and with a thousand more, thousand more demons are hit in the forefront of that war. [3-26-31]
AGAIN in the Yuddha Kand Sarga 21 when the ocean refuses to heed to Rama's request to give a passage to Lanka he uses such force that even the Lord of the OCean Varun is forced to come forward with his hands folded in placation.
ते ज्वलन्तो महावेगास् तेजसा सायक उत्तमाः |
प्रविशन्ति समुद्रस्य सलिलम् त्रस्त पन्नगम् || ६-२१-२७ ततो वेगः समुद्रस्य सनक्र मकरो महान् |
स बभूव महाघोरः समारुत रवस् तदा || ६-२१-२८ महाऊर्मि माला विततः शन्ख शुक्ति समाकुलः |
सधूम परिवृत्त ऊर्मिः सहसा अभून् महाउदधिः || ६-२१-२९
Those excellent arrows having a great speed and ablaze by their splendor, penetrated the waters of the ocean, striking the water- snakes with terror. The great jerk of water with fishes and alligators of the ocean, together with the resonance of the wind became very much dreadful. Immediately that great ocean, became shaken by the cluster of huge waves enveloped by a multitude of conches in its rolling waves and a smoke came out.
These events are such that no mortal human being could have been the perpetrator of and clearly show Rama's divinity.
