TLDR summary:
- There's an argument that taxing consumption rather than income increases incentives to work.
- There's a counter-argument that taxing consumption can increase inequality, particularly by flat-rate consumption tax. (In general, the progressive or regressive nature of the entire tax mix needs to be considered.)
- Both previous arguments (bullets) seem affected (weakened) by what else is done, e.g. how much of the tax money is spent on welfare programs.
The above are economics arguments that are discussed often enough to be considered mainstream. As for pure ideological positions, some the right-wing and/or libertarians in the US attack a consumption tax as a "wealth tax". I'm not sure if anyone has bothered to refute this (poorly framed, IMO) ideological argument. No wealth is taxed through a consumption tax unless it is actually consumed.
This question seems to apply squarely to EU countries, which have both income tax and a fairly high VAT. Actually,
The spread of Value Added Tax (also called Goods and Services Tax – GST) has been the most
important development in taxation over the last half-century. Limited to less than ten countries in the late
1960s it has now been implemented by about 136 countries; and in these countries (including OECD
member countries) it typically accounts for one-fifth of total tax revenue. The recognised capacity of VAT
to raise revenue in a neutral and transparent manner drew all OECD member countries (except the United
States) to adopt this broad based consumption tax. Its neutrality of principle towards international trade
also made it the preferred alternative to customs duties in the context of trade liberalisation.
As for the theoretical arguments for a consumption tax, from the large body of economics papers on this:
The debate on possible consequences of a tax shift from income towards consumption centers around two issues. First, according to standard economic theory,
such a tax shift might be favorable with respect to employment as a consequence
of lower marginal tax rates on labor income, implying higher incentives to take up
work. Second, higher consumption taxes are often associated with lower tax progressivity and higher levels of inequality. However, employment increases from a tax
shift may outweigh adverse distributional impacts. The degree to which there exists
a trade-off between equity and efficiency in this context is an empirical question.
But do note the immediate counter argument on potential inequality increase (that's what regressive tax means). Frankly, if one considers the EU with its typically higher unemployment than the US, I'm not this theoretical argument of consumption tax encouraging work holds much water in practice... Nevertheless, it's actually part of the official EU thinking:
In the recent years, some new aspects of the EU tax policies have
emerged. The Commission invites to increase “quality of taxation” which means
that tax system should generate a proper amount of public revenue and cause
minimal harm to economic growth (EU Commission, 2011abc). One of the
aspects of taxation system quality issue is a modification of tax structure. That
means optimal and efficient allocation of tax burden across various tax subjects.
The Commission invites to improve taxation through “more growth-friendly tax
structure”, which means shifting tax burden from “labor to consumption”
(European Commission (EU Commission, 2011c, p.4-5)
A Canadian parliament website provides a fairly similar argument:
One benefit of most consumption taxes is their limited adverse impact on economic growth. When the income of individuals and businesses exceeds the amount of their consumption, the difference can be invested. For example, investments can be made in the construction of productive assets, which can increase future production and economic growth. A consumption tax encourages individuals and firms to consume less, which means that they are able to invest more, all other factors remaining the same.
A fixed VAT is recognized as a regressive tax, but its success is owed to how easy it is to have it guarantee income for the government.
a VAT is relatively easy to administer and more economically efficient than an income tax. It’s true that a VAT is regressive but this problem can be addressed by making other, more progressive, changes to taxes or spending.
Supposedly the VAT can and is made progressive in some EU countries by having a lower VAT on foodstuff etc. compared to high-end goods. There are methodological issues in coming with this latter conclusion, so it's not universally accepted.. The aforementioned Canadian website also says:
In Canada, the regressive effects of the GST/HST are mitigated somewhat by the zero-rating or exemption of some products, such as basic groceries, and by the refundable GST/HST tax credit that can be claimed by low-income Canadian taxpayers.
An IMF staff note is also mostly on the same key (but they seem to disagree on the regresiveness issue)
Broad-based
consumption taxes, and particularly the VAT, have the advantage of not discouraging saving
and investment decisions. At the other end of the spectrum, income taxes and social
contributions are thought to have the most adverse effects on growth as they interfere directly
with economic decisions (e.g., labor force participation). Within income taxes, corporate
taxes are typically seen as the most harmful to growth, primarily because they discourage
capital accumulation and productivity improvements, while introducing a bias toward the use
of debt finance. Shifting the composition of the tax system from direct to indirect taxes may
have positive effects on growth, but this may come at the expense of equity. For instance, the VAT is generally regressive in advanced economies—at least when assessed against current income rather than current consumption.
In any case, what you are asking about is called designing the tax structure or mix of direct and indirect taxes. There have been lots and lots of studies what is the best way to do this. What ultimately matters from a redistribution perspective is the overall mix:
The theoretical controversy over the optimal tax mix is also mirrored by the diverging solutions adopted in various OECD countries. Cross-country comparisons reveal striking differences in the tax structure especially when considering effective tax rates. For example, during the period 1970–2001, the ratio of the effective labour to consumption tax rate was as high as 1.89 in the United States and as low as 0.77 in Ireland and 0.78 in Portugal. A similar picture emerges for the effective labour to capital tax ratio. Even among European countries, this ratio ranged from 2.31 in Germany to 0.78 in the United Kingdom. The same holds for the effective consumption to capital tax ratio, which ranged from 2.04 in Finland to 0.52 in the United States. These tax mix differences imply that any cross-country comparison of the inequality effects of effective taxation should be seen in the context of relative tax rates, rather than in relation to single tax rates since the use of the latter will only provide a partial assessment.
There's little doubt however that EU countries (which have VAT) gather more overall taxes relative to their GDP compared to the USA. But also note that Australia has a VAT-equivalent called the GST, and its overall taxes are not far from the US ones.

Here's a part of that graph with indirect taxes also added. The US is in the last place among these (has local/state sales taxes):

I can't find the page where I read this right now but I recall Stiglitz saying he was not particularly worried about the regressive nature of the VAT in Europe because those countries with a high VAT also have more welfare programs. (Which probably also negates the idea that consumption tax alone can lead to more uptake of work. But conversely, one can argue that in state with high welfare, lower consumption taxes would encourage even more people to give up work.) Anyway a newer paper (not by Stiglitz) reiterates roughly the same point:
Clearly, arguments made for both points of view can be valid: a strong reliance on the VAT is relatively regressive, if the alternative is to have a well-functioning progressive income tax. On the other hand, if the VAT indeed serves as a money machine and provides the government with more revenues, these revenues can also be used for financing transfers and the provision of (public) goods that can reduce inequality. The overall impacts of the VAT on inequality are, therefore, ambiguous in theory.
This latter paper also finds that in practice the effects of VAT on inequality have been "benign".
Government, even in its best state, is but a necessary evil; in its worst state, an intolerable one.
~Thomas_Paine