There are various mantras which are used to worship Lord Rudra / Lord Shiva. In Shiva Gita various Gods and sages are seen singing various hymns of Lord Shiva. Shiva Gita is a part of UttarKhanda of Padma Purana which I discuss here.. One can read Shiva Gita here.
In the Shiva Gita, Shiva Pradhurbhavam chapter, when Lord Shiva manifests before Lord Rama, various Gods are seen singing various mantras and hymns in praise of him. It is described there as:
सौन्दर्यसारसंदोहां ददर्श रघुनन्दनः।
बृहद्रथन्तरादीनि सामानि परिगायतः।
अग्रगं गरुडारूढं शंखचक्रगदाधरम।
कालाम्बुदप्रतीकाशं विद्युत्कान्त्या श्रिया युतम।।
जपन्तमेकमनसा रुद्राध्यायं जनार्दनम।
पश्चाच्चतुर्मुखं देवं ब्रह्माणं हंसवाहनम।।
स्तुवन्तं भारतीयुक्तं दीर्घकूर्चं जटाधरम।।
अथर्वशिरसा देवं स्तुवन्तं मुनिमण्डलम।
श्वेताश्वतरमन्त्रेण स्तुवन्तं गिरिजापतिम।
सुवर्णवेत्रहस्ताढ्यं नन्दिनं पुरतः स्थितम।।
दक्षिणे मूषकारूढं गणेशं पर्वतोपमम।
मयूरवाहनारूढमुत्तरे षण्मुखं तथा।।
महाकालं च चण्डेशं पार्श्वयोर्भीषणाकृतिम।
कालाग्निरुद्रं दूरस्थं ज्वलद्दावाग्निसन्निभम।।
त्रिपादं कुटिलाकारं नटद्भृङ्गिरिटिं पुरः।
नानावाहनसंयुक्तं परितो मातृमण्डलम।
तत्र त्रैयम्बकं मन्त्रं जपद्द्विजकदम्बकम।।
गायन्तं वीणया गीतं नृत्यन्तं नारदं दिवि।
नृत्यतो नाट्यनृत्येन रम्भादीनप्सरोगणान।।
गायच्चित्ररथादीनां गन्धर्वाणां कदम्बकम।
कम्बलाश्वतरौ शंभुकर्णभूषणतां गतौ।।
गायन्तौ पन्नगौ गीतं कपालं कम्बलं तथा।
एवं देवसभां दृष्ट्वा कृतार्थो रघुनन्दनः।।
हर्षगद्गदया वाचा स्तुवन्देवं महेश्वरम।
दिव्यनामसहस्रेण प्रणनाम पुनः पुनः।।
Raghunanda there saw the essence of beauty. Riding on their own vehicles and carrying respective weapons were the Digpalas shining in their own lusture in their respective places and singing the Brihad and Rathantara Saman. In the front sitting on the Garuda and with the Sankha Chakra and Gada along with the lustrous Lakshmi was Janardana and singing Sri Rudram with the concenterated mind. In the back was seated Brahma in the swan, singing the Rudra Suktas of four Vedas from the four mouths along with Bharati. Devas and Munis were chanting the Atharvasiras and praising the Lord. God of ocean standing beside Ganga was also signing Svetasvatara hymns in praise of Lord of Girija. Great serpents like Aananta who looked as huge as Kailash mountain were seen singing hymns from Kaivalya Upanishad. Nandi was seen in the front carrying a golden danda in his hands. In the south Ganesha was sitting in his mouse. In the north Shanmukha (Kartikeya) was seen sitting in his peacock. In the left and right side of Paramesvara were seen the Mahakala and Chandes. In a far distance was seen the Kalagni Rudra. Three legged Bhringi was dancing in the front along with the Ganas with various faces. In their respective vehicles were seated the divine mothers. Siddha Vidhyadharas were chanting the Panchakshari Mantra. Kinnars were singing the Divya Rudra Gita. Divine Brahmanas were chanting the Tryambakam mantra. Narada was dancing in joy singing song from his Veena. Gandharvas like ChitraRathas were singing. All the Gods of heaven, heavenly snakes etc.. were seen all around singing the songs of Maheswara. Rama became ecstasic on seeing this beautiful scene. Then he started chanting Maheswara Sahasranama with many salutations.
Similarly in this chapter of Anushashana Parva also various Gods are singing various hymns in praise of Mahadeva:
अस्तुवन्विविधैः स्तोत्रैर्महादेवं सुरास्तदा ।
ब्रह्मा भवं तदा स्तुन्वन्रथन्तरमुदीरयन् ।।
ज्येष्ठसाम्ना च देवेशं जगौ नारायणस्तदा ।
गृणञ्शक्रः परं ब्रह्म शतरुद्रीयमुत्तमम् ।।
ब्रह्मा नारायणश्चैव देवराजश्च कौशिकः ।
अशोभन्त महात्मानस्र्त्रयस्र्त्रय इवाग्नय ।।
The deities were engaged in singing the praises of Mahadeva by uttering diverse hymns. The Grandsire Brahma uttering a Rathantara, praised Mahadeva. Narayana also, uttering the Jyestha Saman, sang the praises of Bhava. Sakra also did the same with the aid of those foremost of Vedic Mantras, viz., the Sata-Rudriyam. Verily, Brahma and Narayana and Sakra,--those three high-souled deities,--shone there like three sacrificial fires.
Similarly in chapter 64 of Linga Purana, Parashara also worships Lord Shiva using various hymns like:
अदृश्यंति वशिष्ठं च प्रणम्यारुन्धतीं तत ।
कृत्वैकलिंगं क्षणिकं पांसुना मुनिसन्निधौ ।।
संपूज्य शिवसूक्तेन त्र्यंबकेन शुभेन च।
जप्त्वा त्वरितरुद्रं च शिवसंकल्पमेव च।।
नीलरुद्रं च शाक्तेयस्तथा रुद्रं च शोभनम्।
वामीयं पवमानं च पंचब्रह्म तथैव च।।
होतारं लिंगसूक्तं च अथर्वशिर एव च।
अष्टांगमर्घ्यं रुद्राय दत्वाभ्यर्च्य यथाविधि।। (Linga Purana chapter 64)
Then Parashara bowed in reverence to Adryasanti - his mother, Vasistha and Arundhati - his grandparents. In presence of the sage Vasistha, he made a temporary Iinga of clay. Then he reciting the hymns from the Shiva Sukta, Tryambakam mantra, Tvarita Rudra, Siva Sankalpa, Nila Rudram, Shakteya- Rudra, Vamiya, Pavaman Sukta, PanchaBrahman Sukta, Linga-Sukta and Atharvasiras mantras, adored the Shivalinga. After worshipping the linga appropriately, he offered Ashtangya Arghya to Rudra.
Similarly Srimad Bhagvatam 8.7.29 states:
मुखानि पञ्चोपनिषदस् तवेश
यत् तच् छिवाख्यं परमात्म-तत्त्वं
देव स्वयं-ज्योतिर् अवस्थितिस् ते (S.B. 8.7.29)
O lord, the five important Vedic mantras are represented by your five faces, from which the thirty-eight most celebrated Vedic mantras have been generated. Your Lordship, being celebrated as Lord Śiva, is self-illuminated. You are directly situated as the supreme truth, known as Paramātmā.
Similarly in the Bhakti Yoga chapter of Shiva Gita, Lord Shiva himself states about various hymns which pleases him:
यो वेदाध्ययनं यज्ञअं दानानि विविधानि च।
मदर्पणधिया कुर्यात्स मे भक्तः स मे प्रियः।।
नर्यभस्म समादाय विशुद्धं श्रोत्रियालयात ।
अग्निरित्यादिभिर्मन्त्रैरभिमन्त्र्य यथाविधि ।।
उद्धूलयति गात्राणि तेन चार्चति मामपि ।
तस्मात्परतरा भक्तिर्मम राम न विद्यते ।।
सर्वदा शिरसा कण्ठे रुद्राक्षान्धारयेत्तु यः ।
पञ्चाक्षरीजपरतः स मे भक्तः स मे प्रियः ।।
भस्मच्छन्नो भस्मशायी सर्वदा विजितेन्द्रियः ।
यस्तु रुद्रं जपेन्नित्यं चिन्तयेन्मामनन्यधीः ।।
स तेनैव च देहेन शिवः संजायते स्वयम ।
जपेद्यो रुद्रसूक्तानि तथाथर्वशिरः परम ।।
कैवल्योपनिषत्सूक्तं श्वेताश्वतरमेव च ।
ततः परतरो भक्तो मम लोके न विद्यते ।।
ईशावास्यादि मन्त्रान यो जपेन्नित्यं ममाग्रतः ।
मत्सायुज्यमवाप्नोति मम लोके महीयते ।।
He who studies the Vedas, does Yajnas and Danas as per shastra, and offera his action to me, he is my favourite devotee. He who applies holy ash in his body reciting the mantras like "Agniriti Bhasma.. " no one is dear to me as him. He who wears Rudraksha Maala on his head and neck, and he who always recites Panchkashari Mantra, both these type of devotees are very dear to me. He who applies the holy ash, sleeps on holy ash and conquered all senses, reads the Sri Rudram hymn with concenterated mind, he is Shiva himself from his body. He who chants the Rudra Suktas, chants Atharvasiras, Kaivalya and Svetasvatara Upanishad, no one is superior to him. He who regularly chants the mantras of Isavasya Upanishad is sure to get Sayujya Mukti on me.
So, the above refrences from Padma Purana, Mahabharata, Linga Purana and Srimad Bhagvatam forms the summary of various mantras and hymns used for the worship of Lord Rudra.
Some information of above mentioned mantras and shlokas are as:
1) Panchakshari Mantra:
Considered as one of the most potent mantra of Lord Shiva. Mantra present in YajurVeda Samhita. I discuss Panchakshari Mantra in my answer here and here
2) Trayambakam Mantra:
The "Trayambakam Yajamahe... " is one of the most famous mantra and also known as Mahamrityunjaya or Mokshyadayini Mantra. It is present in RigVeda and YajurVeda and is asking for Moksha from Lord Shiva. I discuss Trayambakam Mantra in my answer here and here.
3) PanchaBrahman Mantras:
PanchaBrahman mantras are from Taittariya Aranyaka of Yajurveda and describe Sadyojata to Ishana forms of Lord Shiva. I discuss PanchaBrahman Mantras in my answer here. These are the same mantras which is stated by Srimad Bhagvatam in 8.7.29. The 38 mantras which are derived from these 5 mantras stated by Srimad Bhagvatam are given by Prabhupada in commentary of 8.7.31:
(1) tat puruṣāya vidmahe śāntyai, (2) mahā-devāya dhīmahi vidyāyai, (3) tan no rudraḥ pratiṣṭhāyai, (4) pracodayāt dhṛtyai, (5) aghorebhyas tamā … , (6) atha ghorebhyo mohā … , (7) aghorebhyo rakṣā … , (8) aghoratarebhyo nidrā … , (9) sarvebhyaḥ sarva-vyādhyai, (10) sarva-sarvebhyo mṛtyave, (11) namas te ’stu kṣudhā … , (12) rudra-rūpebhyas tṛṣṇā … , (13) vāmadevāya rajā … , (14) jyeṣṭhāya svāhā … , (15) śreṣṭhāya ratyai, (16) rudrāya kalyāṇyai, (17) kālāya kāmā … , (18) kala-vikaraṇāya sandhinyai, (19) bala-vikaraṇāya kriyā … , (20) balāya vṛddhyai, (21) balacchāyā … , (22) pramathanāya dhātryai, (23) sarva-bhūta-damanāya bhrāmaṇyai, (24) manaḥ-śoṣiṇyai, (25) unmanāya jvarā … , (26) sadyojātaṁ prapadyāmi siddhyai, (27) sadyojātāya vai namaḥ ṛddhyai, (28) bhave dityai, (29) abhave lakṣmyai, (30) nātibhave medhā … , (31) bhajasva māṁ kāntyai, (32) bhava svadhā … , (33) udbhavāya prabhā … , (34) īśānaḥ sarva-vidyānāṁ śaśinyai, (35) īśvaraḥ sarva-bhūtānām abhaya-dā … , (36) brahmādhipatir brahmaṇodhipatir brahman brahmeṣṭa-dā … , (37) śivo me astu marīcyai, (38) sadāśivaḥ jvālinyai.
4) Sri Rudram:
Sri Rudram is avaliable in all Shakhas of YajurVeda as I discuss in my answer here. The Jabala and Kaivalya Upanishad recommend chanting of Sri Rudram hymn for Moksha/Immortality. One can read Sri Rudram hymn here and with translation here
5) Linga Sukta:
Linga Suktas are from the Taittariya Aranyaka of YajurVeda. They are just before the PanchaBrahman mantras described above. I discuss mantras of Linga Suktas in my answer here. The mantras are also present in MahaNarayaniya and are as:
निधनपतये नमः । निधनपतान्तिकाय नमः ।
ऊर्ध्वाय नमः । ऊर्ध्वलिङ्गाय नमः ।
हिरण्याय नमः । हिरण्यलिङ्गाय नमः ।
सुवर्णाय नमः । सुवर्णलिङ्गाय नमः ।
दिव्याय नमः । दिव्यलिङ्गाय नमः ।
भवाय नमः। भवलिङ्गाय नमः ।
शर्वाय नमः । शर्वलिङ्गाय नमः ।
शिवाय नमः । शिवलिङ्गाय नमः ।
ज्वलाय नमः । ज्वललिङ्गाय नमः ।
आत्माय नमः । आत्मलिङ्गाय नमः ।
परमाय नमः । परमलिङ्गाय नमः ।
एतत्सोमस्य सूर्यस्य सर्वलिङ्ग स्थापयति पाणिमन्त्रं
पवित्रम् ॥ १॥
Skambha Sukta of Atharvaveda also describes a pillar/Skambha/Linga which supports the entire creation.
6) Atharvasiras Mantras:
Atharvasiras Upanishad is a part of Atharvaveda. It is an authentic Upanishad and quoted in numerous old texts like Gautama Dharmasutras19.12, Baudhayana Dharmasutra verse 3.10.10, Vasistha Dharmasutras verse 22.9, etc.. Atharvasiras explains the meaning of various names of Lord Shiva like Rudra, Maheswara, Ishana etc.. and it contains various mantras in praise of Lord Shiva. It also recommends to perform a holy rite called PashuPat Vrata for liberation (which I discuss here). One can read Atharvasiras in Sanskrit here.
7) Ishavasya Mantras:
Isha Upanishad is one of the Mukhya Upanishad and is 40th chapter of Shukla YajurVeda. It was heard by Rishi Dadhichi. The first verse of the Upanishad states "ईशा वास्यमिदं सर्वं" ie. "The entire universe is covered by Isha." One can read Isha Upanishad in Sanskrit here.
8) Svetasvatara Mantras:
Svetasvatara Upanishad is also considered among the Mukhya Upanishad and also quoted in Brahma Sutras. It states "Eko hi Rudro Na Dwitiyaya... " ie. "Rudra is the only thing that exists, no second thing exists other than Rudra." It also contains other names of Lord Rudra like Shiva, Hara, Isha, Ishana, Maheswara, etc... Svetasvatara Upanishad in Sanskrit is avaliable here.
9) Kaivalya Upanishad:
Kaivalya Upanishad is also one of the important Upanishad. While commenting on the name 'Shiva' in Vishnu Sahasranama Adi Shankara has cited a verse from Kaivalya Upanishad as "स ब्रह्मा स शिवः सेन्द्रः सोऽक्षरः परमः स्वराट् ।" Kaivalya Upanishad also contains mantras to glorify Lord Shiva. One of the important mantra which I discuss here contained in this Upanishad is:
उमासहायं परमेश्वरं प्रभुं त्रिलोचनं नीलकण्ठं प्रशान्तम् ।
ध्यात्वा मुनिर्गच्छति भूतयोनिं समस्तसाक्षिं तमसः परस्तात् ॥ 1.7॥
Meditating on the highest Lord, allied to Uma, powerful, three-eyed, blue-necked, and tranquil, the holy man reaches Him who is the source of all, the witness of all and is beyond darkness (i.e. Avidya).
10) Nila Rudriyam:
Nila Rudra Upanishad is a part of Kanda 14 Sukta 3 of Atharvaveda Paipplada Samhita. It contains numerous mantras which glorifies Lord Shiva. One can read NilaRudra Upanishad in Sanskrit here
11) Maheswara Sahasranama:
Sahasranama of Mahadeva is present in various scriptures like Linga Purana, Shiva Purana, Skandha Purana, Anushashana Parva of Mahabharata etc.. Shiva Sahasranama of Anushasana Parva is avaliable here