Today we will look at the form वासिष्ठाः mNp from श्रीमद्-वाल्मीकि-रामायणम् 1.57.14.

अशक्यमिति चाप्युक्तो वसिष्ठेन महात्मना । प्रत्याख्यातो वसिष्ठेन स ययौ दक्षिणां दिशम् ∥ १-५७-१३ ∥
ततस्तत्कर्मसिद्ध्यर्थं पुत्रांस्तस्य गतो नृपः । वासिष्ठा दीर्घतपसस्तपो यत्र हि तेपिरे ∥ १-५७-१४ ∥
त्रिशङ्कुस्तु महातेजाः शतं परमभास्वरम् । वसिष्ठपुत्रान् ददृशे तप्यमानान् मनस्विनः ∥ १-५७-१५ ∥

Gita Press translation – “He was, however, told by the mighty Vasiṣṭha that it was not possible to conduct such a sacrifice. Repulsed by Vasiṣṭha, he proceeded to the southern quarter (13). For the accomplishment of the said purpose the king then sought Vasiṣṭha’s sons at the place where they were actually leading an austere life, engaged as they were in a prolonged course of austerities (14). The exceptionally glorious Triśaṅku beheld (from a distance) the hundred high-minded and supremely resplendant sons of Vasiṣṭha practicing austerities (15).

वसिष्ठस्यापत्यम् (पुमान्) = वासिष्ठः – a (male) descendant of the sage Vasiṣṭha. Here it refers to the son of Vasiṣṭha.
In the verses the विवक्षा is प्रथमा-बहुवचनम्। Hence the form is वासिष्ठाः।

(1) वसिष्ठ ङस् + अण् । By 4-1-114 ऋष्यन्धकवृष्णिकुरुभ्यश्च – To denote the sense of अपत्यम् (descendant) the तद्धित: affix ‘अण्’ may be applied optionally following a syntactically related पदम् in which the सन्धिः operations have been performed, provided the पदम् ends in a sixth case affix and has as its base the name of a sage, or of one born in the ‘अन्धक’, ‘वृष्णि’ or ‘कुरु’ dynasty.
Note: First the सूत्रम् 4-1-83 प्राग्दीव्यतोऽण् prescribes the default affix ‘अण्’ which is over-ruled by the affix ‘इञ्’ (prescribed by the सूत्रम् 4-1-95 अत इञ्) and finally the सूत्रम् 4-1-114 ऋष्यन्धकवृष्णिकुरुभ्यश्च re-prescribes the affix ‘अण्’ because ‘वसिष्ठ’ denotes the name of a sage.
As per the सूत्रम् 4-1-92 तस्यापत्यम् – Following a syntactically related पदम् in which the सन्धिः operations have been performed and which ends in a sixth case affix the तद्धिता: affixes already prescribed (by the prior rules 4-1-83 प्राग्दीव्यतोऽण् etc), as well as those that are going to be prescribed (by the following rules 4-1-95 अत इञ् etc), may be optionally applied to denote the sense of अपत्यम् (descendant.)

(2) वसिष्ठ ङस् + अ । अनुबन्ध-लोपः by 1-3-3 हलन्त्यम्, 1-3-9 तस्य लोपः।

Note: ‘वसिष्ठ ङस् + अ’ gets the designation प्रातिपदिकम्‌ by 1-2-46 कृत्तद्धितसमासाश्च। This allows 2-4-71 सुपो धातुप्रातिपदिकयोः to apply in the next step.

(3) वसिष्ठ + अ । By 2-4-71 सुपो धातुप्रातिपदिकयोः – A सुँप् affix takes a लुक् elision when it is a part of a धातुः or a प्रातिपदिकम्।

Note: The affix ‘अण्’ is a णित् (has the letter ‘ण्’ as इत्)। This allows 7-2-117 तद्धितेष्वचामादेः to apply in the next step.

(4) वासिष्ठ + अ । By 7-2-117 तद्धितेष्वचामादेः – The first vowel among the vowels of a अङ्गम् which is followed by a तद्धित: affix (ref. 4-1-76 तद्धिताः) which is either a ञित् (has the letter ‘ञ्’ as a इत्) or a णित् (has the letter ‘ण्’ as a इत्) takes the वृद्धि: (ref. 1-1-1 वृद्धिरादैच्) substitute.

Note: The अङ्गम् ‘वासिष्ठ’ gets the भ-सञ्ज्ञा by 1-4-18 यचि भम् which allows 6-4-148 यस्येति च to apply in the next step.

(5) वासिष्ठ् + अ । By 6-4-148 यस्येति च – When the letter ‘ई’ or a ‘तद्धित’ affix follows, the (ending) इ-वर्ण: (letter ‘इ’ or ‘ई’) or the (ending) अवर्ण: (letter ‘अ’ or ‘आ’) of the अङ्गम् with the भ-सञ्ज्ञा takes elision (लोपः)।

= वासिष्ठ । The प्रातिपदिकम् ‘वासिष्ठ’ declines like राम-शब्दः।

(6) वासिष्ठ + जस् । By 4-1-2 स्वौजसमौट्छष्टाभ्याम्भिस्ङेभ्याम्भ्यस्ङसिभ्याम्भ्यस्ङसोसाम्ङ्योस्सुप्‌।

(7) वासिष्ठ + अस् । अनुबन्ध-लोपः by 1-3-7 चुटू and 1-3-9 तस्य लोपः। 1-3-4 न विभक्तौ तुस्माः prevents the ending letter ‘स्’ of ‘जस्’ from getting the इत्-सञ्ज्ञा ।

(8) वासिष्ठास् । By 6-1-102 प्रथमयोः पूर्वसवर्णः – When a अक् letter is followed by a vowel (अच्) of the first (nominative) or second (accusative) case then for the two of them (अक् + अच्) there is a single substitute which is the elongated form of the first member (the अक् letter.)

(9) वासिष्ठाः । रुँत्व-विसर्गौ by 8-2-66 ससजुषो रुः and 8-3-15 खरवसानयोर्विसर्जनीयः।

We can similarly derive the following –
१) विश्वामित्रस्यापत्यम् (पुमान्) = वैश्वामित्रः – a (male) descendant of the sage Viśwāmitra