As a teaching assistant, how important are student evaluations? I have seen a number of résumés of doctoral students, but only a few of them listed their evaluation scores when presenting their teaching experience. 


How important are these scores in evaluating the teaching capabilities of a student? 
How does one ensure that the students are sincere in their evaluations?


In case a professor does a slipshod work of a course, not teaching in depth or cramming a lot of syllabus in a short time, there is only so much a TA could do to salvage the course for the students. 


How does the TA make the best of a bad job in such a situation? 
Apart from holding weekly office hours and lenient grading(!), what is the maximum a TA can do, after all?
This is an excellent question, for faculty as well as students!


How important are these points in evaluating the teaching capabilities of a student?


Obviously this varies significantly in different departments and institutions, but in my experience, the scores themselves are not that important.

My department does pay attention to these numbers when allocating future TAships, but definitely not in isolation.  Narrative reviews from the instructors carry significant weight.  The people doing the assignments also know which courses are unpopular, and which instructors are irresponsible, and adjust the evaluation accordingly, at least in principle.  In practice, there are only three evaluations: (1) truly outstanding TAs, who are considered for teaching awards; (2) truly abysmal TAs, who are not rehired, at least without retraining (and since we have a TAship requirement, this has teeth); and (3) everyone else.

When we evaluate tenure-track faculty candidates, teaching ability is usually a second-order concern, but it is a concern.  Poor evaluations on an applicant's CV are a red flag—why didn't they just omit them?  Good evaluations are mostly a signal to look further.  Teaching awards carry more weight.  Recommendation letters that directly praise the applicant's teaching ability — with concrete and credible details — are even better.  Similar issues arise when evaluating faculty for tenure, with one big difference: omitting the teaching scores is not an option.


How does one ensure that the students are sincere in their evaluation?


You can't.  Sorry.

However, I believe you can increase the fraction of sincere (and positive) responses by consistently treating your students with respect.  Make your expectations clear from day one, and enforce them consistently.  Invite feedback throughout the semester, and respond to it quickly and appropriately.  Apologize quickly for mistakes, thank students publicly for useful suggestions, but do not buckle on high standards.  Give timely, consistent, and useful feedback on coursework.  Above all, do not waste your students' time; the correlation between hard work and low evaluations is much higher if the students don't see any benefit to doing the work.

&lt;Insert standard confirmation bias warning here.&gt;


How does the TA make the best of a bad job [if the instructor is irresponsible]?


First, do your own job as best as you can.

Second, raise your concerns with the instructor; be respectful but brutally honest.  If the instructor is unresponsive, raise your concerns with your instructor's boss; be respectful but brutally honest.  (Note: Disagreement is not the same as being unresponsive.)  If your instructor's boss is also unresponsive, your department doesn't really care about poor teaching; they're likely to ignore your evaluations, even if they are low.


Apart from holding weekly office hours and lenient grading(!), what is the maximum a TA can do, after all?


There are many more things that TAs can do.  At a minimum, hold office hours that the students actually find useful; don't just show up.  Distribute practice problems, and offer feedback on the students' solutions.  Hold weekly review/discussion/problem-solving sessions.  As aeismail suggests, write review notes.  If the instructor covered too much, distill down their main points; if the instructor didn't cover enough, expand on the key ideas they missed.  Offer to give a few guest lectures, and then give fantastic guest lectures.

More self-servingly: Make sure the students see you working to overcome your instructor's shortcomings.  If the students don't see you fighting on their behalf (even if you are), they'll write you off as yet another useless academic, like your instructor.  But if you can make them believe you're on their side, they'll reward you.  I think this is why students often reward "lenient grading"; if the students think the coursework is a waste of time, they'll see lenient graders as their allies.

Obviously this all takes time.  As aeismail says, TAs usually have many other responsibilities, especially to their own classes, projects, research, families, and sanity.  It is frighteningly easy for committed and caring TAs to find themselves being abused by less committed instructors (or even departments).  Set limits.