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1. Ensure your linkedin profile is upto date.

2. put 2-3 posts in your interest area (can be basic legal updates as well)

3. connect with people you want to intern under (PAs/SAs/Partners of firms or junior associates of chambers you want to work under)

4. You can use this message if you want "Hello Sir,

It is great connecting with you. As a young law student, I am currently looking for interning opportunities. It will be of great help if you could share the email id of ***, wherein I can forward my application for the same."
^ this is what i used, and the person was kind enough to grant me an internship spot.

Feel free to modify it, to make it more amiable(?) for your profile.

Hope this helps.

Alternatively, if you have recently published an article/research paper, which is in your domain area, try sharing it with the abovementioned on linkedin. this can help in breaking ice. and then ask for an internship.

there is a higher chance that someone might respond to you seeking their feedback/advice vis a vi directly an internship.

Good luck!
As an A0 (batch of 23) who is currently working at a T1 who keeps getting such requests, please note -

1. A0's themselves are on probation and new to the firm who they do not have any definitive say in recruitment of interns.

2. While messaging please make your message with some achivement that normally is not done by other students eg an empirical study or a research grant or a good paper publication etc. Ensure that it is relevant to the practice area you are applying to.

3. It's always better to reach out to Partners/Counsels/ PA's/SA's who have more leeway.

A0's like me would honestly love to refer good applications on LinkedIn but please understand our hands are also tied.

All the best!