Data representation of largest DCT coefficients In many examples of data compression, we take the S largest coefficients of a signal in a sparse basis (eg. 1% of largest DCT coefficients) and set the rest to zero.

How do we practically represent the compressed signal? In the above example, once the smallest coefficients have been set to zero, the signal is directly converted back into the time domain and compared to the original. What if we wanted to store the data?

In other words, how do we concisely store the indexes of the zeroed out coefficients?
Your question is very accurate. Storing the largest  (1 % for instance) coefficients only from a sparsifying transform (DCT, wavelet, else) is fool's-gold, since you (more importantly the decoder) don't you where those are located. 

And storing their binary indices counterbalances the compression gain for the sparsification. So trade-off should be found between theoretical sparsity and storage. For images, in JPEG, a low-frequency to high-frequency parsing scheme (zig-zag) is used. After quantization, lots of zeroed-out coefficients are compacted with End-Of-block symbols.  In JPEG 2000, a multi-scale and bit-wise  path is often  used, and zeroed-out coefficients are compacted with EZM-SPIHT like coding, that perform sort of vector coding of zero bit-plane.

So, some pre-determined way of parsing coefficients, based on prior knowledge on the signal or image structure, is often useful for actual compression, and applicable to specific class of data.