In the Srimad Valmiki Ramayana, there are two Rakshasas by the name of Trishiras, one being the commander-in-chief of Khara’s army and the other being Ravana’s son.
1. Trishiras, a chief in Khara’s army
This Trishira is introduced in the 23rd Sarga of the Aranyakanda, while describing the army of Khara that proceeds towards Rama, consequent to Shurpanakha’s insult:
महाकपालः स्थूलाक्षः प्रमाथिस्त्रिशिरास्तथा।
चत्वार एते सेनाग्रे दूषणं पृष्ठतोऽन्वयुः॥३-२३-३३
Mahakapaala, Sthulaaksha, Pramaathi, Trishira, these four were at the front of the army and they followed Dushana, marching behind him.
After destroying the 14000 Rakshasa army, only Khara and Trishiras were left, where Trishiras takes Khara’s permission to fight with Rama. Their battle is described in the 27th Sarga of the Aranyakanda, where ultimately Rama severs the three heads of the Rakshasa:
सायकैश्चाप्रमेयात्मा सामर्षस्तस्य रक्षसः॥१७॥
शिरांस्यपातयत् त्रीणि वेगवद्भिस्त्रिभिः शरैः। स धूम शोणितोद्गारी रामबाणभिपीडितः॥१८॥
न्यपतत् पतितैः पूर्वे समरस्थो निशाचरः। हत शेषास्ततो भग्ना राक्षसाः खरसंश्रयाः॥१९॥
And with three sharp and rapid arrows, and even with his own exasperation, Rama of inestimable aptitude rolled the three heads that demon. When that nightwalker is highly tortured with Rama's arrows he collapsed spewing blood with fumes, whereat he is standing in war, at where his heads rolled just now.
The killing of Trishiras alongwith Khara and Dushana also finds mention in the Ramacharitmanas during Shurpankha’s lament to Ravana:
खर दूषन तिसिरा कर घाता। सुनि दससीस जरे सब गाता॥ अरण्यकाण्ड दोहा २२
Hearing that Khara, Dushana and Trishira had been killed, Ravana’s body burned.
2. Trishiras, Ravana’s son
He is one of the seven sons of Ravana and actively participates in the war, after Kumbhakarna’s death. He boosts a lamenting Ravana’s morale by himself offering to head into the war, alongwith his brothers - Devantaka, Narantaka and Atikaya, as described in the 69th Sarga of the Yuddhakanda. He engages with Angada, but ultimately meets his end at the hands of Hanuman, who chops his three heads with a sword, as described in the 70th Sarga of the Yuddhakanda:
तेन मुष्टिप्रहारेण संचुकोप महाकपिः। कुपितश्च निजग्राह किरीटे राक्षसर्षभम्॥४६॥
स तस्य शीर्षाण्यसिना शितेन किरीटजुष्टानि सकुण्डलानि।
क्रुद्धः प्रचिच्छेद सुतोऽनिलस्य त्वष्टुः सुतस्येव शिरांसि शक्रः॥४७॥
Hanuman became enraged with the blow of Trishiras's fist and caught the crowned head of the demon. Then just as Indra had chopped the three heads of Tvashta’s son Vishvarupa, Hanuman chopped off the three heads of the demon, furnished with diadems and ear-rings.
Thus we can see that there were two Trishiras in the Ramayana. For more information on Tvashta’s son Vishvarupa, this answer can be checked.
Note:- wherever the main Sanskrit text and translation, as per the linked website were different from that of Gita Press Gorakhpur, the Gita Press Gorakhpur version has been preferred (though the translations (wherever discrepant) are in blockquotes they represent an English translation of the Gita Press Hindi text by me)