Hello!
I have personally found this to be really task-dependant, rather than there being a single "right" level of detail that applies to all research tasks or written assessments. I've split my thoughts on this between research tasks and written assessments, as I think they differ slightly.
Research Tasks
Some research tasks that I have been given have been very tightly scoped (e.g. "answer these four questions and don't go beyond that"), where the expectation is clearly to be concise and targeted. However, others have been much broader, asking for general research on an area of law or a commercial trend, where a more comprehensive overview is actually expected. Therefore, the level of depth (+ time spent on the task) can vary quite a lot depending on what you have been asked to do by your supervisor.
I've also found that the same thing applies to things such as footnotes and sources. Different supervisors tend to have different preferences - for example, some I've worked with didn't mind how sources were presented, while others were quite particular (e.g. preferring hyperlinks embedded in the text rather than formal footnotes, or vice versa). Because of this, I would always recommend asking these kinds of questions when the task is first given to you. This shows a proactive attitude and that you are thinking about how best to deliver the work, and most supervisors will appreciate that and tell you exactly what they want from the task.
If they don't specify, I would just take a common-sense approach and focus on what is most clear and easy for them to digest. Personally, I tend to include sources in any event (even if not explicitly asked), as it makes your work more transparent and useful - they can quickly check where something has come from or dig deeper if needed.
In terms of structure, something that always worked for me was to make the main document reasonably comprehensive (unless I'm told otherwise) and pair it with a short executive summary at the top of the email. This means that they can immediately see the key answers or conclusions (especially if they are busy), but still have the detail underneath if they want to go into it. This balance between clarity and depth was something that I received good feedback on, so I think it is useful to include if you have the time!
On the partner vs client point, I haven't personally done any research tasks directly for clients - it has almost always been for a supervisor who then uses that research to write an email to a client (or other client-related tasks). Therefore, I would generally always include sources and be slightly more thorough, unless you are told to be concise. Your supervisor can always strip things back for the client, but it is harder and more time consuming for them to include additional key details or sources if they are missing.
Written Tasks
For written tasks, I would approach things slightly differently depending on who the instructions state you are writing for. If it is client-facing, I would aim to be much more concise, clear, and to the point than if you are writing a research task. I found that having a strong structure really helps with this - things such as clear headings, a logical flow and signposting mean that you can cover the key points without discussing lots of unnecessary detail. I haven't previously had to include formal "sources" in a written exercise, but I would always refer to the relevant information within the text so that it is clear what your point is based on.
For client-facing written work specifically, I would focus on what actually matters to the client and their interests. This could be things like the practical implications, key risks, and clear answers to any specific questions that they have - rather than trying to include everything you have found. Depth is still important in a client-focused task, but you should be selective and purposeful in the areas that you discuss, rather than trying to discuss everything.
Overall, I think the safest approach is to clarify the task instructions where you can, as that is often what your supervisor would be looking for. However, if they leave it open-ended or don't specify, I'd always aim for something that is clear, well-structured, and easy to navigate - ultimately, you are trying to make less work for them, so it should be set out in the most efficient way!
I hope that helps, and best of luck!!