Ashadha Masa Mahatmya

Astrologically, the positioning of celestial bodies during Ashadha is said to be unfavourable for initiating new endeavours or events. It is a time for spiritual practices, reflection, and focusing on inner growth rather than external activities that are considered auspicious.

This is the month associated with the Vamana Avatara of Vishnu. 
This month is the starting of many four month long vratas and anushtanas ( chaturmasya)

Dakshinayana - Karka Sankranti

This is the first month in Dakshinayana. The day of the sankranti is auspicious. 

Ashadha Shukla ekadashi - Deva-Shayani Ekadashi
 Lord Vishnu falls asleep in the milky ocean ( kshira Sagara) on Shesha Naaga on this day. That is why it is called Shayani ( sleep). He awakens from his sleep four months later on Prabodhini Ekadashi in Kartika masa. This period is known as Chaturmasya "four months".

On this day, being awake the whole night and worshipping lord Vishnu and Lakshmi is mentioned in many scriptures. ( like we worship shiva during shiva-ratri) 

आषाढशुक्लपक्षे तु कामिका हरिवासरः २४।
तस्यामेका च मूर्तिर्मे बलिमाश्रित्य तिष्ठति ।
द्वितीया शेषपृष्ठे वै क्षीरसागरमध्यतः २५।
स्वपित्येव महाराज यावदागामि कार्तिकी ।

Chaturmasya vrata

Chaturmasya vrata sankalpas (Whatever vratas one is inspired to perform in the chaturmasya)
is taken on the evening ( sandhya) of Ashadha shukla dvadashi. 

many difficult vratas are taken up by sadhakas during chaturmasya. The ones mentioned in the scripture are really tough in the modern times, but still the followers perform it diligently with great devotion. 
The theme is around making a sacrifice to please the lord. By giving a physical representation to the sacrifice like, "I will give up eating certain foods" or "I will sleep on the floor as my lord is also in sleep". the sadhaka is reaffirming the inner willingness to surrender to any hardship, any test. There is a hope for reward at the end of the sacrifice in case of "Kamya" vratas. For example " I am performing yoga-abhyasa for four months and wish to get deeper knowledge of brahman".  

The traditional practices are as follows - 
Sadhaka sleeps on floor 

Shakavrata
Sacrifices eating vegetables in shravana, curd in bhadrapada, milk in ashvayuja, split grains/grams in kartika.

Those who cannot perform the shaka vrata can take sankalpa of yoga-abhyasa, eating havishya-anna. Give up eating meat during chaturmasya etc. 

स च लोके मम सदा श्वपचोऽपि प्रियंकरः ३१।
चातुर्मास्यं नयंतीह ते नरा मम वल्लभाः ३२।
चातुर्मास्ये हरौ सुप्ते भूमिशायी भवेन्नरः ।
श्रावणे वर्जयेच्छाकं दधि भाद्रपदे तथा ३३।
दुग्धमाश्वयुजि त्याज्यं कार्तिके द्विदलं त्यजेत् ।
अथवा ब्रह्मचर्यस्थः स याति परमां गतिम् ३४।
एकादश्या व्रतेनैव पुमान्पापैर्विमुच्यते ।
कर्तव्या सर्वदा राजन्विस्मर्तव्या न कर्हिचित् ३५।
शयनी बोधिनी मध्ये या कृष्णैकादशीभवेत् ।
सैवोपोष्या गृहस्थस्य नान्या कृष्णा कदाचन ३६।

Ashadha Shukla Poornima - 

This vrata is mostly performed by women. On this day Prajapati Daksha started his 12 year long Satra ( type of Yagnya ) in which Shiva was not invited and Sati immolated herself. After which she was in the form of a "Kokila" bird before she was reborn as Parvati and reached Shiva once again. This vrata starts on Ashadha poornima and ends on Shravana poornima which is also Raksha-bandhana this period is a month long.

Details of this vrata 
This is a month long vrata where one must observe Vak shuddhi - purity of speech, Shareera shuddhi - purity of the body and  Sarva bhuta daya - compassion towards all living beings (Abstaining from inflicting harm physical and verbally towards other beings).

The Kokila bird had the ability to contain Sati's spirit until her rebirth was facilitated. Hence the bird symbolises great purity, strength and determination. In a kamya manner, the vrata can be performed for protection, prosperity, progeny, a consort as the kokila's (Sati's) aim was to reach Shiva. 
If performed without any kamana, depending on the intensity of the vrata, it will result in the awakening of the inner voice, Vak shuddhi, Vak shakti, and health.

आषाढ़पौर्णमास्यान्तु सन्ध्याकाले ह्युपस्थिते । सङ्कल्पयेन्मासमेकं श्रावणीप्रभृतिह्यहम् । स्नानं करिष्ये नियता ब्रह्मचर्य्ये स्थिता सती । भोक्ष्यामि नक्तं भूशय्याङ्करिष्ये प्राणिनान्द-याम् । इति सङ्कल्प्य पुरुषो नारी वा ब्राह्मणान्तिके ।
प्राप्यानुज्ञान्ततः प्राह्णे सर्वसामग्रिसंयुतः” इत्यादि ।

Sanyasis take chaturmasya - vasa - sankalpa ( staying in one place for four months) perform vyasa pooja on this day. 

2. Vyasa Pooja ( Guru Poornima) - Celebrated in surrender and reverence to the Guru tatva, it is also considered the birthday of Veda Vyasa.  

3. Gopadma Vrata (Four month long)

In this vrata, all the devata kalpana is done in the divine cow ( Go). There are mantras to invoke various deities in various parts of the cow and then offer them prayers. 
Later 33 times the motif of "Gopadma" of the cow are drawn in a sacred place ( in the go shala, in temple, at home in pooja room, near tulasi plant). Prayers are offered to Vishnu / Shiva / Ishtadevata. This is done everyday until kartika shukla ekadashi. A special naivedyam ( Apoopa / Appam / Fritter like thing made of rice flour and jaggery) at the end of the vrata is offered to lord and distributed to worthy people.

This vrata was instructed in the puranas in many contexts, in one context even Krishna prescribes this for his sister Subhadra. 

आषाढस्य तु पूर्णायां गोपद्मव्रतमुच्यते ॥
चतुर्भुजं महाकायं जांबूनदसमप्रभम् ॥ १२४-१३ ॥
शंखचक्रगदापद्मरमागरुडशोभितम् ॥
सेवितं मुनिभिर्देवैर्यक्षगंधर्वकिन्नरैः ॥ १२४-१४ ॥
एवंविधं हरिं तत्र स्नात्वा पूजां समाचरेत् ॥
पौरुषेणैव सूक्तेन गंधाद्यैरुपचारकैः ॥ १२४-१५ ॥
आचार्यं वस्त्रभूषाद्यैस्तोषयेत्स्निग्धमानसः ॥
भोजयेन्मिष्टपक्वान्नैर्द्विजानन्यांश्च शक्तितः ॥ १२४-१६ ॥
एवं कृत्वा व्रतं विप्र प्रसादात्कमलापतेः ॥
ऐहिकामुष्मिकान्कामांल्लभते नात्र संशयः ॥ १२४-१७ ॥

Ashada Krishna ekadashi - yogini ekadashi 
This ekadashi is performed for getting relief from diseases. There is a story associated with it. Once there was a Yaksha whose task was to bring special flowers for Sri Hari's pooka that was  performed by Kubera. One day the Yaksha ( Hema-Mali) gets carried away by the beauty of a yakshi and visits her with the flowers, kubera waits for a long time and then discovers that Hema-mali got distracted, he curses him to suffer on earth as a leper. During that birth somehow Hema-mali reaches the ashrama of Markandeya Rishi and the Rishi instructs him to fast on Yogini ekadashi and he is relieved of his curse.

आषाढस्यासिते पक्षे योगिनी नाम नामतः ।
एकादशी नृपश्रेष्ठ महापातकनाशिनी ३।
संसारार्णवमग्नानां पोतभूता सनातनी ।
जगत्त्रये सारभूता योगिनी व्रतकारिणाम् ४।

Ashadha Amavasya 
In different parts of the country, this amavasya is celebrated for the well-being of the husband. Many observe the Pati-sanjeevani vrata on this day. The details of the vrata are tradition based.
( I have not yet come across the puranic reference to this Vrata, nor have I found it in  other works like dharma kaustubha etc, when I find it, I will update this section) 

Reference - 

कोकिलाव्रतमप्यत्र प्रोक्तं तद्विधिरुच्यते ॥
पूर्णिमायां समारभ्य व्रतं स्नायाद्बहिर्जले ॥ १२४-१८ ॥
पूर्णांतं श्रावणे मासि गौरीरूपां च कोकिलाम् ॥
स्वर्णपक्षां रत्ननेत्रां प्रवालमुखपंकजाम् ॥ १२४-१९ ॥
कस्तूरीवर्णसंयुक्तामुत्पन्नां नंदने वने ॥
चूतचंपकवृक्षस्थां कलगीतनिनादिनीम् ॥ १२४-२० ॥
चिंतयेत्पार्वतीं देवीं कोकिलारूपधारिणीम् ॥
गंधाद्यैः प्रत्यहं प्रार्च्चेल्लिखितां वर्णकैः पटे ॥ १२४-२१ ॥
ततो व्रतांते हैमीं वा तिलपिष्टमयीं द्विज ॥
दद्याद्विप्राय मंत्रेण भक्त्या सस्वर्णदक्षिणाम् ॥ १२४-२२ ॥
देवीं चैत्ररथोत्पन्ने कोकिले हरवल्लभे ॥
संपूज्य दत्ता विप्राय सर्वसौख्यकरी भव ॥ १२४-२३ ॥
द्विजं सुवासिनीस्त्रिंशदेकां वा भोजयेत्ततः ॥
वस्त्रादिदक्षिणां शक्त्या दत्वा नत्वा विसर्जयेत् ॥ १२४-२४ ॥
एवं या कुरुते नारी कोकिलाव्रतमुत्तम् ॥
सा लभेत्सुखसौभाग्यं सप्तजन्मसुनारद ॥ १२४-२५ ॥

इति श्रीभविष्ये महापुराण उत्तरपर्वणि श्रीकृष्णयुधिष्ठिरसंवादे कोकिलाव्रत नामैकादशोऽध्यायः ।। ११ ।।

 Reference - Gopadma Vrata