Making the selection transparent in QGIS 3.0

Make selection color transparent by reducing opacity in Set Selection Color window.

Settings menu > Options > Canvas & Legend > Selection color

The same setting is available on a project-specific basis in the Project > Properties menu. You may need to restart QGIS to see the changes.

Here are another few options for making the background visible behind selected features:

Use an outline symbol type:

You can pair this with centroid fill:

Or add multiple, offset outlines for a more filled polygon appearance. The example image below has 6 symbol layers with offsets of 0, 1.5, 3, 4.5, 6, and 7.5 mm.

In QGIS 3.10.9, this might not work. Using this path (Settings > Options > Canvas & Legend > Default Map Appearance) to change colors/transparency of selected features will be overridden by the project properties. I successfully changed color and transparency of selected features by going to Project > Properties > General > Selection Color and making my changes there.