It depends on purpose. Print, Web, display on a small digital frame, or on a huge (80inch for example) screen.
The big issue is a misunderstanding of DPI and PPI, they are not the same thing! Also saving at 72 was good when we had CRT monitors. These days most monitors are around 100 so saving at 72 might just be a bit low
DPI and PPI are often assumed to be the same and thus often interplaced with each other, when in reality they are quite different.
Dots Per Inch (DPI) is a printing term and is not really relevant to digital photography, it is not a photographic term, it is a printing term. DPI is a measure of how many tiny, tiny droplets of ink a printer can lay-down in its dither pattern to form one inch of a print. You will often see an inkjet printer marketed as 1440 DPI (can be as low as 720 DPI and as high as 5760 DPI). You tell the printer which DPI mode to print when you select options just before you print. So to summarise, DPI is a printing term.
Pixels Per Inch (PPI) is the key term for digital images. It is a description of the number of pixels from your original image (X pixels by Y pixels - height by width) that will be used to tell the printer to print one inch on paper. Assuming a
sharp original shot with good technique the higher the PPI, the better the quality print you can achieve - this is testably true even well beyond most claims of 360 PPI being the most you need ... 600 PPI images can easily be seen to be much sharper again if this data is available at good quality from the original file).
PPI is a logical term - changing the PPI of a particular file does not in any way affect the file itself - it is simply a decision about how many pixels of the available pixels you will use to print an inch on page. You can choose any number you like - from 1 to infinity. The de facto standard for high quality, photographic printed images is 300 PPI - that is, for each inch of the printed image, there must be 300 source pixels to use.
So DPI relates to how many little droplets of ink are used by a printer, PPI relates to the resolution of the digital image.
Now with that, you should be able to work out what you want/need for your images.