Lawn Care
•    Sow or Fertilize warm-season grasses like Bermuda, Zoysia, and St. Augustine.
•    Do not fertilize cool-season grasses like tall fescue, Kentucky bluegrass, and perennial ryegrass.
•    Tall fescue and bluegrass should be mowed at a height of 2 ½ to 3 ½ inches.

Fertilizing
•    Long season vegetables should be side-dressed 6 to 8 weeks after germination.
•    Fertilize summer flowers as well.
•    Feed hybrid rhododendrons and azaleas with an acid fertilizer.

Planting
•    Vegetable to be planted in May: snap beans, pole beans, bush lima beans, cantaloupe, sweet corn, cucumbers, eggplant, okra, southern peas, sweet peppers, hot peppers, pumpkins, squash, tomatoes, and watermelons.
•    Gladioli bulbs can be planted.
•    Summer annuals like begonia and geranium can also be planted.

Pruning
•    Hybrid rhododendrons and azaleas can be pruned as soon as they finish blooming.
•    Outgrown hedges can be pruned.
•    Pinch off garden mums till mid-July.
•    Pick off any azalea leaf galls that may form.
•    Spring bulb foliage should not be cut back till yellowing of foliage occurs.

Spraying 
•    Rhododendrons should be monitored for borers and sprayed if present.
•    Spray for iris borers in iris beds
•    Scout and spray for bagworms on shrubs and trees.
•    Use recommended herbicide to control poison ivy and honeysuckle if desired.
•    Start spraying squash vines to prevent borers.
•    Spray blueberry plants for blueberry borers.
•    Start fungicide treatment on tomatoes, which show signs of blight.
•    Continue with rose spraying program.
•    Continue fungicide program for fruit trees and bunch grapes.
(Pesticides should be used sparingly!  Use only when needed and always follow the label)

Other Activities
•    Observe landscape shrubs for azalea-lace bug, boxwood-leaf miner, euonymus-scale, hemlock, juniper-spruce mites, and hybrid rhododendron-borer.
•    Observe broccoli, cabbage, and cauliflower, for worms and spray if needed.
•    Consider putting down mulch in beds to help with weed control.

From: http://www.ces.ncsu.edu/Gardening+Calendar+May/

See also: https://durhammastergardeners.wordpress.com/2013/05/04/what-to-do-in-may/,  https://durhammastergardeners.wordpress.com/2012/04/26/what-to-do-in-the-garden-in-may/