Only wild magic effects that specifically cast spells can be managed with these methods.
Looking at the Wild Magic Surge table (PHB, P.104), there are generally four categories of effects that can be rolled:
(1) Effects that cause you to cast a spell. Examples include fireball, confusion, fog cloud, fly, etc. Since you are casting a spell, these can be counterspelled by you or another spellcaster. When these spells have a duration (such as fog cloud, or mirror image), they can also be ended with dispel magic.
(2) Effects that cause an ongoing effect. Examples include changing your height, growing you an extra eye, you (or other creatures) becoming invisible, etc. Since this is not a spell it can't be ended by dispel magic. Some effects indicate how they can be removed (for example, the skin color change on a 23-24 roll), but beyond that, the wild magic surge effect can't be ended.
(3) Effects that cause an instantaneous non-spell effect. Examples include dealing lightning damage to all creatures near you, teleporting you, or restoring all of your sorcery points. Since these aren't spells or an ongoing effects, they can't be counterspelled or ended with dispel magic.
(4) Effects that summon a creature. Examples include a modron, flumphs, or a unicorn. Summoned creatures generally can't be dismissed with dispel magic.
So why can't dispel magic help, especially for cases like category (2)? Here's the spell wording:
Choose one creature, object, or magical effect within range. Any spell of 3rd level or lower on the target ends.
And here's a very important clarification from Sage Advice:
Dispel magic has a particular purpose: to break other spells. It has no effect on a vampire’s Charm ability or any other magical effect that isn’t a spell.
By my count, of the 50 different effects on the Wild Magic Surge table, nine of them can be counterspelled, and seven of those (all except fireball and magic missile) have durations and can be dispelled.