Its contemporary, Kharoshthi was used in Gandhara. Many scholars consider the Ashokan inscriptions recorded on rocks, stone pillars, excavated caves and stone slabs to be the earliest Brahmi records.
The spread of Buddhism and Hinduism to South-east, East and Central Asia made Indian scripts and Sanskrit to develop in these nations.
This made Sanskrit as an International language, was used extensively in official, religious records and Literature. Many of these scripts are now obsolete.
Hence, a revival is required on these vulnerable writing systems which in turn can preserve Indian literature and heritage.
Presently Sanskrit is written in all major Indian scripts except Tamil and Gurmukhi in which this language is not natively written. But, since 19th century Devanagari has became a de facto and official script for Sanskrit language at national and international level.
But at regional level vernacular scripts are still prevalent. Many of these scripts are used to write local languages,while some of them are used for Sanskrit only!
There is script whose name is Sanskrit.
Most of scripts are written from left to right similar to Devanagari and Latin. But , Sanskrit is also written from Right to Left in Kharoshthi script similar to Arabic. In Mongolian Galik version Sanskrit is written from Top to bottom.
All the script mentioned here are Brahmic( Indic scripts) . But Latin, Mongolian are one of the non-Brahmic scripts. Sanskrit is written in more than 40 scripts. This makes Sanskrit , a language written in largest number of scripts in the world!
But there is no single place or website which provides entire list of scripts in which Sanskrit language is written. This is an attempt based on my knowledge and research.
Brief description of these scripts
Major Indian scripts are namely Devanagari, Bengali, Gujarati, Oriya, Kannada, Telugu and Malayalam which are widely used by the people in religious, ceremonial and scholarly aspects.All these scripts have complete repertoire of letters used in Sanskrit.
Other South Asian scripts
Grantha script was (is) widely used in Tamil region to write Sanskrit as writing in Tamil script was not suitable. The modern Malayalam, Tigalari, Sinhala and southeast scripts such as Thai, Javanese etc. are derived from this script.
Sharada was the principal literary script used in Kashmir used to Sanskrit and other north Indian regions . A large number of manuscripts and literary works exist in this script.Modern Gurmukhi, Takri and Landa scripts are evolved from Sharada.
Ranjana script is one of the beautiful and calligraphic script primarily used in Nepal. A variation of this script called Lantsha, Wartu is widely used in Tibet, Himalayas and parts of China, Mongolia as a ornamental script in Buddhism.
Newar also known as Nepal script, Newari, Prachalit Nepal is used write Nepal Bhasha and Sanskrit.This script is similar to Bengali and Devanagari.
Tirhuta also known as Maithili was used in Mithila region of Bihar and Nepal to write Maithili language and Sanskrit. A large no of Panjikas and manuscripts are available.
Tigalari script is used in Malnad and coastal regions of Karnataka, a derivative of Grantha. This was used only to write Sanskrit. Manuscripts of Veda, Upanishads, Jyotisha and other literary works are available.
Sinhala is used to write Sinhala and Sanskrit in Sri Lanka. Many Buddhist works are written in this script.
Nandinagari used in South India mainly in Karnataka to write Sanskrit only, derived from Nagari. Many copper plates, manuscripts, coins of Vijayanagara kingdom are discovered.
Siddham a buddhist script developed from northern indian branch, but present day mainly used in Japan, China and Korea used to write Sanskrit only. It is a highly calligraphic and aesthetic script. It is also known as Siddha-matrika.
Bhaikshuki was used around the 11th and 12th centuries CE in eastern India and parts of Tibet, Nepal. Very few manuscripts and attestations are so far available.
Kharoshthi an ancient script written from right to left was used for Sanskrit and Gandhari. It was in use from the middle of the 3rd century BC until it died out in its homeland around the 3rd century CE in present day Afghanistan and Pakistan.
Tamil and Gurmukhi(Punjabi) do not contain complete repertoire of Sanskrit letters. Hence, they are rarely used for Sanskrit. Even when written they are modified to write Sanskrit.
Central Asian scripts
Tibetan, a buddhist script derived from northern branch of Brahmi used in Vajrayana Buddhism to write religious Sanskrit works in Tibet,Nepal,Himalayas and Mongolia.
Phags-Pa, a script written vertically.It is derived from Tibetan used in Mongolia and Tibeta to write various languages inc. Sanskrit.
Soyombo and Zanabazar Sqaure script are influenced from Tibetan and developed in Mongolia.
Sanskrit is also written in a non-Brahmic script- Mongolian(Galik).
Several historic script derived from Central Asian Brahmi are Tocharian, Khotanese, Tumshuqese and so-called Uighur Brahmi are collectively known as Turkestani used to write Sanskrit and vernacular languages.
South-east Asian scripts
Many scripts of insular South-east Asia used in Theravada Buddhism along with Pali include Burmese(Myanmar), Thai, Khmer, Tai Tham etc.
The scripts of Indonesia are Javanese and Balinese where there is a strong presence of Indian tradition
.
Due to Globalisation Sanskrit is now widely written and transliterated using Latin script with modification namely IAST.
Other minor scripts of India include Modi, Kaithi, Saurashtra, (Jaina-nagari) and many more.
It is worth mentioning here that many historical variants or stages of the scripts are not listed here such as Pallava, Gupta, Kushana, Chalukya etc.
Awaiting for reply and suggestions.
List of scripts