This is a controversial question ... I shall have a go at it.
The terms nutritional status and anthropometric status are often used interchangeably. Nutritional status refers to the internal state of an individual as it relates to the availability and utilisation of nutrients at the cellular level. This state cannot be observed directly so observable indicators are used instead. There are a range of observable indicators (biochemical, clinical, and anthropometric) of nutritional status, none of which taken alone or in combination are capable of providing a full picture of an individual's nutritional status. There is, therefore, no single “gold-standard” indicator of nutritional status.
Nutritional status can be usefully defined at the individual, as opposed to the cellular, level as the ratio of nutrient reserves (muscle and fat) to the nutrient requirements of organs (brain, liver, heart, kidneys, lungs, &c.). It is generally recognised that muscle plays a special role as a nutrient reserve during infection and that infection is a major etiological factor in acute undernutrition (wasting). W/H expresses the relationship between weight and height. In children, about 4% of weight is nutrient reserves in muscle. About 96% of weight is, therefore, unrelated to nutrient reserves. Height is almost completely unrelated to the nutrient requirements of organs. MUAC, however, is directly related to muscle mass and is, therefore, a direct measure of nutrient reserves.
The limited evidence that is currently available suggests that an index known as the lean-mass ratio (LMR), the ratio of the estimated mass of the limbs to the estimated mass of the trunk, is the best anthropometric indicator of nutritional status. The available evidence suggests that MUAC uncorrected for age or height is a better indicator of nutritional status than all other practical indicators and that weight-for-height is not associated with LMR and is the worst practical indicator of nutritional status.
An alternative to examining the association between an anthropometric indicator and "nutritional status" is to examine the prognostic or predictive value (i.e. of predicting death) of various indicators. When this has been done, W/H has been consistently shown to be least effective predictor of mortality and that, at high specificities, MUAC is superior to both height-for-age and weight-for-age.
In terms of indicators that are practical to collect in developmental and emergency settings, MUAC has the best claim to be a practicable “gold-standard” of nutritional status.
BMI is a species of weight-for-height. It is weight divided by height squared. Since sitting to standing height ratio changes as a child grows (i.e. the infant body shape is lost and limbs become longer) it is sensible to adjust BMI for age. This should make BMI-for-age a better indicator than weight-for-height. There are, however, some issues. The first is that it is a complicated measure requiring height, weight, and age to be collected and calculations and look-up in tables. This means that it is expensive and time-consuming to measure and cannot be used for mass frequent screening particular by CHWs or even in primary healthcare centres (height boards are not part of essential clinic equipment packs and height measurement is not part of IMCI). Also, age is notoriously difficult to collect with accuracy in many settings. All of the "-for-age" indicators (e.g. H/A, W/A, MUAC/A) are known to be very sensitive to errors in ascertaining age (i.e. small differences in age make large differences to calculated indicator values) and, in cross-sectional applications, errors in age tend to increase with increasing age. This does not invalidate "-for-age" indicators in applications such a growth monitoring in which age is ascertained early and, with repeated measures over time, error will decreases with increasing age. This means (e.g.) that the use of W/A in growth monitoring programs is sensible (when coupled with MUAC it has proved very strong and creates a useful linkage between GMP and CMAM programs).
The real answer to this question is "It depends what you want to use it for, where you want to use it, and how much you are willing to spend to get the measure". My "bias" is towards child survival programming in poor countries. For this type of programming there is really no choice. It has to be MUAC. W/H and BMI/A are, in these programs and settings, damaging (i.e. to program effectiveness via coverage) distractions. The use of W/H in these contexts is (IMO) a historical mistake and should be dropped.
Just my tuppence.