What should I choose IIT Roorkee Microelectronics and VLSI or NIT Trichy VLSI?

Well, here is something that you can do to get a better sense of career prospects in those areas after M.Tech from IITs/NITs

First search for VLSI on LinkedIn dot com (I am not able to post the url as you will be asked to login first) and add filter for school names, mentioning several IITs and NITs.

Now check-out the profile of people who are getting listed and see if they have done M.Tech and which company they are working in or have worked in past for VLSI. I got some results from 1st page itself:

1) NIT Karnataka, Presently working in Open Silicon, Inc.

2) IIT Kanpur, Atmel Corporation

3) IIT Delhi, CVC Ltd

4) IIT Madras, Wipro

5) IIt Kanpur, Wipro

With this limited research, I can say two things:

1) There are good career opportunities for VLSI

2) There is good career progression for people who did M.Tech from VLSI from IITs / NITs - from the search result I found people moving to senior designations quickly.

I also know the companies who have hired M.Tech in VLSI from NITs and IITs. Now the next question is, which of these companies visited IITs in last 2-3 years - that should be available on the websites of the colleges. I can also connect with some of the people I found and simply ask them.