It is crucial you use the correct plug AND screw size AND drill size. Your packet of wall plugs should tell you all this. Select the type of plug you need. That will tell you on the pack what size drill bit and screw to use with it.

Use a good quality branded plug supplier with a range of plugs for different uses. That way you can be confident that the plugs will do the job you want reliably. I use Plasplugs range of plugs.

Pictured below from the Plasplugs website are the main types I use. These are available from most good DIY shops. Checkout their website for more info. and more to choose from for different wall types.
www.plasplugs.com

Concrete and brick regular duty. Use No. 8 screw and 6mm masonry drill

Plasterboard regular duty. Use No. 8 screw with 10 mm drill

Plasterboard heavy duty. Use No. 8 screw with 10 mm drill