from my perspective, engineer-Doctor or singer can be anyone from vaidic society, as i read some books like Mahabharat(Aranya parva, shanti parva), Bhagvad Geeta, Meluha(set of books who is best to educate our vedic cast system as per my brief's).
now let's take some notes here.
A son of Brahmin, if he fails to complete his education, becomes a
Shudra. Similarly, son of a Shudra or even a Dasyu, if he completes his
education can become a Brahmin, Kshatriya or Vaishya. This is pure
meritocracy. The way degrees are accorded today, Yajnopaveet was
provided in Vedic system. Further, non-compliance with the code of
conduct for each Varna could result in taking away of the Yajnopaveet.
for example, Parsuram, by born he was Bramhin but as per their karma can you put them in category of bramhin as they demolished kshatriya form whole world several times..
so, what is the actual Varna system in our Vedic time?
In Vedic culture, everyone is considered to be born as Shudra. Then
based on his or her education, one becomes a Brahmin, Kshatriya or
Vaishya. This completion of education is considered to be a second
birth. Hence these three Varnas are called “Dwija” or twice-born. But
those who remain uneducated for whatever reasons are not discarded from
society. They continue as Shudra and perform support-activities for the
Another example MR. Ravan, who is by born Bramhin, a great shiv bhakt and pandit and .. and lots more but actually he is considered as RAKSHASH (forgive me lord Ravan). so in another word a varna vyavastha is also depends on what person think, what he want to be.
but for only your Question perspectives,
An engineer should be in category of Bramahin, why? because Lord Vishvakarma is bramhin.
A singer is Vaishya, for example Apsara's,
and a doctor is also a bramhin , Lord Dhanvantri..
The original varna system was quite flexible in which one’s varna could be changed based on one’s skill and was not fixed as is often understood.
There is no element of birth-based discrimination of any manner in the Vedas.