mjhjmtu

The horizontal Bhasma for Siva worshipers and the vertical marking for Vishnu worshipers are well known. I have also seen Smarta folks wearing a vertical mark that is different from the ones worn by Vaishnavites.

Do these markings have a basis in scripture?

I am providing first the scriptural basis for the Tilak worn by the Shiva devotees.

The God S’iva is always hidden under the cover of ashes; so the
S’aivas, the devotees of S’iva are to use the Tripundra. The Brâhmanas
are to use daily this Tripundra. Brahmâ is the Prime Brâhmin. When He
used Tripundra on His forehead, what need to tell, then, that every
Brâhman ought always to use it! Never fail, out of error, to besmear
your body with the ashes as prescribed in the Vedas and worship the
S’iva Lingam. The Sannyâsins are to apply Tripundra on their forehead,
arms, chest, uttering the Triyamvaka mantra with Om prefixed and also
the five lettered mantra of S'iva “Om Namah S’ivâya.” The Brahmâchâris
should use Tripundra of ashes, obtained from their own fire, uttering
the mantra “Triyâyusam Jamadagneh,” etc., or the mantra “Medhâvî”,
etc. The S’ûdras in the service of the Brâhmins are to use the ashes
with devotion, with the mantra “Namah S’ivâya.” The other ordinary
persons can use the Tripundra without any mantra. To besmear the body
all over with ashes and to put on the Tripundra is the essence of all
Dharma; therefore this should be used always. The ashes from the
Agnihotra Sacrifice or from Virajâgni (Virajâ fire) are to be
carefully placed on a clean and pure basin. Cleansing hands and feet,
one is to sip (perform Âchamana) twice, and then, taking the ashes in
the hand, utter the five Brahmâ mantras “Sadyoyâtam prapadyâmi,” etc.,
and perform short Prânâyâma thrice; he is, then, to utter the seven
mantras “Fire is ashes,” “water is ashes,” “earth is ashes” “Teja is
ashes,” “wind is ashes,” “ether is ashes,” “All this whatsoever is
ashes” and purify and impregnate the ashes with the mantra by blowing
out air through the mouth. Then one is to think of Mahâ Deva,
repeating the mantra “Om Apojyoti,” etc., and apply dry ashes of white
colour all over the body and become sinless. After this he is to
meditate on the Mahâ Visnu, the Lord of the universe and on the Lord
of the waters and repeat again the mantras “Fire is ashes” and mix
water with the ashes. He is, then, to think of S’iva and apply ashes
on his forehead. He is to think of the ashes as S’iva Himself and
then, with mantras appropriate to his own Âs'rama (stages of life) use
the Tripundra on his forehead, chest and shoulders. By the middle
finger and ringfinger he is to draw the two lines of the ashes from
the left to the right and by his thumb draw a third line of ashes from
the right to the left. These Tripundras are to be used in the morning,
midday and in the evening.

Source - Devi Bhagavatam, Book 11, Chapter 9.

Further details can be found in the following verses which elaborately describe how to wear the Tilakas and what Mantra to use at the various steps.

First perform the water bath, then the bath of the ashes. In case one
cannot have the water bath, one is to have the ash bath. Washing the
hands, feet and head with the mantra “Is'ânah Sarvavidyânâm,” etc.,
and uttering “Tatpurusa” one is to besmear one's face with ashes and
by the mantra “Aghora” apply ashes on one's chest; with the mantra
Vâmadeva. He is to use ashes on his navel; and with the mantra “Sadyo
Jâta,” etc., all over his body; he is to quit his former cloth and put
on another fresh cloth. Wash your hands and feet and sip (do Âchaman).
It will serve the purpose if one simply uses Tripundrak and if one
does not besmear the whole body with ashes. Before the midday one is
to use Bhasma with water; but after the midday with dry ashes one is
to draw the Tripundra lines of ashes with the forefinger, middlefinger
and ringfinger. The head, forehead, ears, neck, heart, and the arms
are the places whereon the Tripundras are used. On the head the ashes
are applied with five fingers and with the mantra “Haum”; on the
forehead, the Tripundra is applied with mantra Svâhâ by the
forefinger, mindlefinger and ringfinger; on the right ear, it is
applied with “Sadyojâta” mantra; on the left ear, with “Vamadeva”
Mantra; on the neck with Aghora mantra by the middlefinger; on the
chest with “Namah” mantra by the forefing middlefinger and ringfinger;
on the right arm with vasat mantra by the three fingers; on the left
arm with “Ham” mantra by the three fingers; and on the navel, the
ashes are to be applied with the mantra Îs'ânah sarva devânâm by the
middlefinger. The first line in every Tripundra is Brahmâ; the second
line is Visnu; and the third, the topmost line is Mahâ Deva. The line
of ashes that is marked by one finger is Îs'vara. The head is the
place of Brahmâ; the forehead is the site of Îs'vara; the two ears are
the seats of the two As'vins and the neck is where Ganes'a resides.The Ksattriyas,
Vais'yas,and S’ûdras are to use Tripundras without any mantra; they
are also not to use the ashes on the whole of the body. The lowest
classes (e.g. the chândâlas, etc.,) and the uninitiated persons are to
use the Tripundraks without any mantra.

Source - Book 11, Chapter 11 of the same Purana.

Needless to repeat that Bhashma or ashes is used for the Shaiva Tilaka. And, the Tilaka is named Tripundra.

Greatness of wearing the Tripundra is the content of Book 11's 12th Chapter which can be read from here.

UPDATE:

The U-shaped Vaishnava Tilak is known as the UrdhaPundra.

The same Purana also describes about the procedure of wearing it and about it's greatness:

76-84. Since that day the Devas did not allow any other devotee of
S’iva to go to the newly created hell Kumbhîpâka. Thus I have
described to you the excellent greatness of the Bhasma. O Muni! What
more can there be than the glories of the Bhasma! O Best of Munis! Now
I am telling you of the usage of Ûrdhapundra (the vertical marks)
according to the proper province of the devotees. Listen. I will now
speak what I have ascertained from the study of the Vaisnava S’âstras,
the measure of Ûrdhapundra, according to the Anguli measurements, the
colour, mantra, Devatâ and the fruits thereof. Hear. The earth
required is to be seen red from the crests of hills, the banks of the
rivers, the place of S’iva (S’iva Ksettram), the ocean beaches, the
ant-hill, or from the roots of the Tulasî plants. The earth is not to
be had from any other places. The black coloured earth brings in
peace, the red-colour earth brings in powers to bring another to one’s
control; the yellow-coloured earth increases prosperity and the
white-coloured earth gives Dharma (religion). If the Ûrdhapundra be
drawn by the thumb, nourishment is obtained; if it be drawn by the
middle finger, longevity is increased; if it be drawn by nameless or
ring finger, food is obtained and if it be drawn by the fore finger,
liberation is attained. So the Ûrdhapundras ought to be drawn by these
fingers, only be careful to see that the nails do not touch at the
time of making the mark. The shape of the Ûrdhapundra (the vertical
mark or sign on the fore-head) is like a flame or like the opening bud
of a lotus, or like the leaf of a bamboo, or like a fish, or like a
tortoise or like a conch-shell. 85-95. The Ûrdhapundra, ten Angulis
high is the super best; nine Angulis high, is best; eight Angulis
high, is good; the middling Ûrdhapundra is of three kinds as it is of
seven Angulas, six Angulas, or five Angulas. The lowest Ûrdhapundra is
again of three kinds as it is four Angulas, three Angulas or two
Angulas high.

Devi Bhagavatam 11.15