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Querying wolfram alpha for the melting point of aspirin (link) returns $pu{140 °C}$. The Wikipedia page for Aspirin lists $pu{136 °C}$ instead, citing this book.

How am I supposed to know which figure to cite? And in general, what should I do when I encounter contradicting chemical data?

Wolfram company doesn't conduct any experimental determinations of physical constants for chemical compounds and uses literature data sources.
The webpage for ChemicalData Source Information lists numerous sources of chemical information used by the company's products, including Wolfram Alpha.

Wolfram Alpha Knowledge Database is linked with 87th ed. (2006) of CRC Handbook of Chemistry and Physics (m.p. $pu{135 °C}$), whereas Wikipedia cites newer 92nd ed. (2011) of CRC Handbook of Chemistry and Physics (m.p. $pu{136 °C}$).

Obviously, Wolfram's sorting algorithm preferred another data source despite it's quoting the CRC Handbook.
Why and how it did that cannot be answered as the server-side engine is proprietary.

Note that even newer 97th ed. (2017) of CRC Handbook of Chemistry and Physics adds measurement error of $pu{4 °C}$ (m.p. $pu{136(4) °C}$).
The difficulty of determining the exact value probably arises from the fact that at that temperature range thermal decomposition already begins, which is reflected in m.p. dependence on heating rate and atmosphere and melt solidifying at lower temperatures (down to $pu{118 °C}$ [1, table 3.4]).

References

1. Aspirin and Related Drugs; Rainsford, K. D., Ed.; CRC Press; Taylor & Francis: London, 2004. ISBN 978-0-7484-0885-6.

The CRC Handbook of Chemistry and Physics ($mathrm{86^{th} Ed.}$, section on Physical Constants of Organic Compounds) provides an MP of $pu{135^circ C}$
for aspirin (2-(Acetyloxy)benzoic acid).

According to the Handbook:

The data in the table have been derived from many sources, including both the primary literature and evaluated compilations.

If you need a highly reliable source you'd want to perform a more thorough search rather than rely on a tertiary (encyclopedic) source such as the wikipedia, and in particular ideally find primary sources of data (original literature reporting experimental results) or perhaps secondary (compilation of scientific results).

This is a pretty good guide to different sources of data:

Sources are considered primary, secondary, or tertiary depending on
the originality of the information presented and their proximity or
how close they are to the source of information. This distinction can
differ between subjects and disciplines. In the sciences, research
findings may be communicated informally between researchers through
email, presented at conferences (primary source), and then, possibly,
published as a journal article or technical report (primary source).
Once published, the information may be commented on by other
researchers (secondary sources), and/or professionally indexed in a
database (secondary sources). Later the information may be summarized
into an encyclopedic or reference book format (tertiary sources).

I would regard the CRC Handbook as a tertiary source of information based on the above definition, but it's generally considered highly reliable. Wolfram Alpha or the wikipedia are also tertiary sources but because of the way they are compiled, unless stated and verified you can't be sure where the data comes from. One of the ongoing efforts at the wikipedia is to include a mechanism to ensure that such information is verified.

As regards the CRC, you should attempt to use the most recently available edition, but the reported melting point of aspirin is not likely to have changed by much in the last hundred years (and yet consider the differences between the values quoted in your question, and this and another answer).