Normal-types aren't typically the most stand-out species in the mainline Pokémon games, but they generally possess attributes that make them sturdy and reliable, as even without the ability to deal super-effective damage they have immunity and resistances to cover them stats-wise. Similarly, there are a wide array of Normal moves that prove ever-reliable in both casual and more competitive settings.
They tend to hit for respectable damage as well as cover for defensive efforts. An added boon is that since the typing is more "generic," it allows for far more flexibility and range in what Pokémon can learn most Normal moves.
Protect
A popular Normal move to run on a wide variety of Pokémon for defensive purposes is Protect. If the move is successfully executed, it shields the user from the vast majority of the opponent's attacks. It's incredibly useful, and as long as the timing is right, it can work particularly well in competitive or casual double-battle settings.
Stage hazards and a few select attacks will bypass it and it has higher chances of backfiring if consecutive uses are attempted, but as long as the player knows their enemy, Protect is an exceptionally versatile defensive move in a variety of situations.
Soft Boiled
There aren't a lot of healing moves in-game compared to damage-dealing and moves that boost Pokémon stats, so having Soft-Boiled to cover this area already bodes well for the Normal-type's competitive viability. It's, unfortunately, a signature move for Chansey and Blissey, but does manage to reach a few other Pokémon in terms of accessibility.
Soft-Boiled heals (only) the user for an amount that's half of the user's max HP, and given the likes of Blissey can use this, it can be nightmarish for the opposition. Blissey is already a defensive wall as is, so being able to execute this can easily frustrate them by prolonging the match. Especially so since this move surprisingly allows for 10 uses (PP) in battle.
Extreme Speed
Extreme Speed is an understandably loved offensive Normal move to use (and a classic synonymous with one of the best Fire-type Pokémon, Arcanine), especially in competitive battling. It's a Physical move, which many Normal-type Pokémon are, and has reliable statistics to support it. Extreme Speed has a very solid 80 Base Power and 100 Percent Accuracy.
Though the biggest boon this move arguably has is that not only is it a Priority move, but as of generation 5 it's been granted a +2 Priority boost. With Speed being a major factor in every competitive fight, this attack will prove incredibly reliable to quickly dispatch high-threat opponents. It understandably has a base PP of 5 uses, as being able to hit that hard that fast could easily be abused otherwise.
Substitute
Another classic Normal move, this time all the way back from the gen one games Red, Blue, Green, and Yellow, is Substitute. There's a risk to using it, but it's another excellent defensive and stalling non-damaging move to use in competitive play. Substitute has the user sacrifice 25 percent of its max HP to create a literal substitute to take damage on the user's behalf.
The substitute lasts until its own HP has been chipped away, allowing the user to prolong the battle and exhaust the target along with the rest of their team. It makes for a proper battle of attrition.
Swords Dance
Swords Dance is often a go-to stat-boosting move regardless of the Type a given Pokémon is. At the very worst, this move is a solid holdover if the player has their eye for a more fitting stat buff for later in the game, and an exceptional mainstay at best.
Physical Attack is one of the most important stats to train in the games, especially for a lot of Normal-type Pokémon, and a single-use of Swords Dance raises the user's Attack by two stages. This saves on prep time for bolstering stats in the middle of a battle, and the accessibility of it makes it an easy slot to fill in on a Pokémon's moveset.
Body Slam
If Extreme Speed is mostly associated with Arcanine, Body Slam typically brings another fan-favorite Pokémon to mind, Snorlax. That's important in this case because Snorlax is a Physical tank, with its offensives geared toward the Physical Attack stat. Body Slam has a hefty 85 Base Attack and 100 percent Accuracy stats, making physically oriented Pokémon benefit greatly from this move.
Even more so with the STAB (Same-Type Attack Bonus) that comes if the user is also a Normal-type, but there are additional potential effects with it. As long as the target isn't in the semi-invulnerable states granted by the likes of Fly and Dig, Body Slam bypasses Accuracy checks, deals double damage if it connects with a Minimize user, and has a 30 percent shot at inflicting Paralysis, a potential nightmare for any glass-cannon Pokémon.
Hyper Beam
Hyper Beam was among the heaviest hitters of the Normal-type move pool since gen 1. This attack was even a common offensive powerhouse that bosses in the game's single-player questline used. Likewise, it can be a good high-power filler move for players to use in-game casually, and is widely accessible.
It's a Special Normal-type move, but with its whopping 150 BP and 90 Accuracy, it's still a killer, namely with a STAB attached. Reasonably enough, though, it comes with just 5 PP uses and requires a recharge after immediate use.
Giga Impact
The fourth generation of games brought Pokémon the direct equivalent to Hyper Beam with Giga Impact. The latter's also a daunting offensive move with the same 150 BP/90 Accuracy spread, along with the same caveat of having to recharge after it's executed.
But this provides a great alternative in that Giga Impact is a Physical move. This gives physically oriented Pokémon, and physical Normal-types especially, a better option to pack the most punch and play to the user's specific statistical strengths. Given how physical a lot of Normal types are, this is good for versatility.
Return
When it comes to versatility and accessibility, Return is arguably the best in both categories. This offensive move has no set BP and comes with a 100 percent Accuracy stat. Damage output, in this case, is calculated by how high the friendship level is between the user Pokémon and the player.
Since the player will ideally be using their Pokémon throughout the course of the single-player story, friendship should inevitably hit its max stat. With the BP having a ceiling of 102 and able to be used across a variety of different species, it's no wonder Return is frequently used with Eevee and final-evolution stage Normal-types. The only downside is that Sword and Shield won't allow it to be used.
Boomburst
Undoubtedly the strongest Normal attack when it comes to raw damage output vs. risk is Boomburst. It has a fantastic 140 BP, 100 Accuracy, and even has 10 base PP uses. The real risks that come with it are that it affects all other Pokémon on the field, and the user can't execute Boomburst for two turns if hit with Throat Chop.
Aside from that, it's a no-brainer Special move to use with the Pokémon compatible with it, like Noivern and Exploud. And, while very exclusive, it gets an incredible 30 percent attack boost if the user has the Punk Rock ability, i.e. Toxtricity.