"I am the lead psychometrician at a personality test publisher, so I will attempt to answer your question.
To begin, it is important to note that no test is "scientifically valid". Validity is not an element of a test, but specifically has to do with test score interpretation. (see the Standards for Educational and Psychological testing 1999, or Messick, 1989). That being said, the Myers Briggs is not a scientifically valid personality assessment. However, personality assessments can be validated for specific purposes.
Moving onto the bigger issue with the Myers-Briggs: Decision consistency. The Myers-Briggs proclaims a reliability (calculated using coefficient alpha) of between .75-.85 on all of its scales (see Myers-Briggs testing manual). These are general, industry standard reliability coefficients(indicating that if you were to retest, you would get a similar score, but not exact). However, the Myers-Briggs makes additional claims about bucketing individuals into 1 of 16 possible personality types. That you can shift up or down a few points if you were to retake the test on any of the four distinct scales means that you may be higher on one scale than another simply through retaking the test due to measurement error. In fact, literature shows that your personality type will change for 50% of individuals simply through retesting. (Cautionary Comments Regarding the Myers-Brigg Type inventory, Consulting Psychology Journal: Practice and research, summer, 2005). This result indicates very low decision consistency. The low decision consistency is also a mathematical inevitability given 16 personality profiles using 4 scales and scale reliability around .8.
Given the low decision consistency, and given that claims the Myers-Briggs makes about about your personality(validity information) depends on the decisions made by the test to be consistent and not subject to change simply based on retesting, it is highly unlikely that there can be a solid validity argument supporting the Myers-Briggs as a personality indicator. Maybe there are studies showing that it can be used in a very specific context, but sweeping generalizations about the tests use are not going carry much weight.
Now, as a working professional in the field, the Myers-Briggs does NOT have a good reputation as being a decent assessment. It has marketed well to school systems and has good name recognizability, but it is not a well developed exam. There are much better personality assessments available, such as SHL's OPQ32 or The Hogan personality inventory. Now, I don't want to say any of these are good. The best correlations between job performance and personality assessments is about .3 (indicating about 9% of the variance in a persons job performance can be accounted for by a personality assessment). That is the BEST personality assessments can do in terms of job performance... and a correlation of .3 is not worth very much (considering that tests like ACT or the SAT can correlate upwards of .7 with first year college GPA under ideal circumstances)."